After considering many different configurations of star systems, I have decided that the most stable system where the two visible stars would always be 180 degrees apart as seen from the planet would have to be an artificially constructed one with an intermediate or supermassive black hole at the center, with the habitable planet orbiting close to the black hole, and a ring of an even number of dim white dwarfs stars plus two mai n equence stars of the same mass orbiting the black hole at a significantly greater distance. See parts seven to twelve.
I consider this a great question, because it asks for something almost but not quite totally impossible, and so I can explain how to make it possible.
Part One: Climate.
You may need someone more expert in calculting the climites of planets to figure out hte climate your planet might have. A planet whichorbits too close to a small dim star might be tidally locked to the star (one side always factng the star) which may have bad effects on the planetary climate.
A planet with constant sunlight might also have climate problems.
Part Two: Relative masses of planets and stars.
One of the first and most obvious facts about orbits is that much less massive objects orbit around much more massive objects. Actually two orbiting objects will both orbit around their barycenter, their common center of gravity. But if one object is much more massive than the other one, the barycenter will be much closer to the more massive one, and often inside the body of the more massive object.
You desire a star system with two stars and one planet along with optional other possible objects which are not required.
If the planet has to be habitable for Earth humans then it should probably have at least 0.25 the mass of the Earth and less than 10 tmes the mass of Earth - quite probably less than 2 times the mass of Earth. A giant planet would not have a solid surface or be habitable for humans. Jupiter, the largest planet in our solar, has 317.8 time the mass of Earth, and many planets in other star sytems are more massive than Jupiter.
Theoretically, the dividing line between the most massive planets and objects called brown dwarfs should be roughly about 13 times the mass of Jupiter, or roughly 4,131.4 times the mass of Earth. The dividing line between the most massive brown dwarfs and the least massive stars should be about 75 or 80 times the mass of Jupiter, or about 23,835 to 25,424 times the mass of Earth.
Since the Sun is about 330,000 times as massive as Earth, the least massive stars would have only about 0.072 to 0.077 the mass of the Sun. The most massive stars know have about 100 times the mass of the Sun, making the most massive stars about 1,390 times as massive as the least massive stars.
However, stars which are capable of having planets which are naturally habitable for humans have a much narrower range of possible masses; say between 0.5 and 1.4 times the mass of the Sun, to be generous. That is about 165,000 to 462,000 times the mass of Earth. So if your planet has to be naturally habitable for humans with a mass of 0.25 to 2.0 that of Earth, in a system with two stars in the range of 165,000 to 462,000 times the mass of Earth, each star will have about 82,500 to 1,848,000 times the mass of the habitable planet.
So the stars can not orbit around the planet. The stars would have to orbit around each other, and the planet would have to orbit around one or both of the stars.
Maybe the planet could stay in the barycenter between the two stars as they orbit each other. As others have written in thier answers, such a configuration would not be stable for geological eras of time. How long would it take for a planet to naturlaly become habitable for humans after the planet formed? If Earth is a good sample, about four billion years, which is far too long for the planet to stay in the Barycenter.
Part Three: Planetary orbits in binary star systems.
So in a binary star system the two stars would orbit around each other. There are two types of possible orbits of planets in a binary star system.
In an S-type or non-circumbinary orbit a planet would orbit around one of the stars at a distance whch was a fraction of the distance between the stars. In a P-type or circumbinary orbit a planet would orbit around both of the stars, at a distance greater than the distance between them.
https://en.wikipedia.org/wiki/Habitability_of_binary_star_systems#:~:text=Habitability%20of%20binary%20star%20systems%20is%20determined%20by,more%20of%20all%20star%20systems%20are%20binary%20systems.
Obviously a planet in a P-type orbit would never be exacly halfway between the two stars. The two stars would always be much less than 180 degrees of arc apart in the sky of the planet. During a planetary day one star would rise, and then later the other star would rise and both stars would be in the sky and then the first star would set and then later the second star would set. Then there would be night until the first star rose again.
You could never have one star rising while the other star was setting.
Continued later.
So what would be needed would be an S-Type or non circumbinary orbit where the planet orbited one of the stars at a fraction of the distance between the two stars.
In non-circumbinary planets, if a planet's distance to its primary exceeds about one fifth of the closest approach of the other star, orbital stability is not guaranteed.2
https://en.wikipedia.org/wiki/Habitability_of_binary_star_systems#:~:text=Habitability%20of%20binary%20star%20systems%20is%20determined%20by,more%20of%20all%20star%20systems%20are%20binary%20systems.
So the closest the two stars get should be at least 5 times the distance betweent the planet and the star. If the two stars had equal masses and luminosities, the nearer star would give the planet at least 25 times as much light and heat as the farther star did.
Suppose that the planet orbits a K2V class star with 0.82 the mass and 0.37 the luminosity of the Sun. Suppose that the farther star is a F2V class star with 1.46 the mass and 5.13 the luminosity of the Sun.
https://en.wikipedia.org/wiki/K-type_main-sequence_star
https://en.wikipedia.org/wiki/F-type_main-sequence_star
If the two stars were at the same distance the F2V star would give the planet 5.13 times the heat and light that the K2V star did. If the F2V star was at least 5 times as far from the planet as the K2V star, it would give the planet no more than 0.2052 times the heat and light that the K2V star did.
If the planet orbited one star, and if the other star had the same mass and luminosity and was at least 5 times as far away. the farther star wold give the planet less than 0.04 times the heat and light that the nearer star did.
And it might not be a problem if the farther star gives the planet much less heat and light than the near star gives it. That would make the days when only the far star was visible much less hot than the days when only the near star was visible. That would give the planet a temperature cycle much more like Earth's cycle alaterantely heatng up in the day and cooling down in the night.
I note that from Earth the Sun has an apparent magnitude of -27 (lower numbers are brighter) and the full moon has an apparent magnitude of -13.
https://en.wikipedia.org/wiki/Magnitude_%28astronomy%29
According to my rough calculations, the Sun is approximately 399,367.08 times as bright as the full moon. So if the near star appeared about as bright as the Sun appeared in the sky of the planet, and the farther star was only 1/631.95, or 0.00158, as bright as the near star, it would still be 631.95 times as bright as the full moon.
I think that the farther star could be less than 1 percent as bright as the nearer stall while still being bright enough to make the sky blue, and thus appear to be a "sun" when rising and setting.
Part Four: Getting the rotational period correct.
If your planet is going to be habitable, it has to have a rotational period and day length compatable with habitability. If the planet rotates too fast and has too short a day, it will become unstable and might fly apart. If the days and nights get too long plants might die during the long nights without light, and the days can get too hot and the nights too cold for life to survive.
Stephen H. Dole, in Habitable Planets for Man, 1964, discusses the conditions necessary for a planet to be habitable for humans 9and for life with similar requirements).
https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf
He discusses planetary rate of rotation on pages 58 to 61. On page 60 he says:
Just what extremes of rotation rates are consistent with habitability is difficult to say. These extremes, however, might be estimated at, say, 96 hours (4 Earth days) at the lower end of the scale and 3 to 3 hours per revolution at the upper end, or at angular velocities where the shape becomes unstable because of the high rotation rate.
So those are limits to consider while choosing a length of day for your planet.
Part Five: Sidereal and solar days.
A sidereal day is the period in which the Earth rotates exactly 360 degrees of arc with respect to the distant stars, but not with respect to the Sun.
A Siderial day is approximately 23 hours, 56 minutes and 4.1 seconds, or 86,164.1 seconds, long.
https://en.wikipedia.org/wiki/Sidereal_time
While orbiting the Sun, the Earth travels almost one degree of arc along its orbit around the Sun each sidereal day. So after one sidereal day the line between the center of the Earth and a spot on the Earth's surface that pointed at the center of the Sun will be pointing in the same direction. But because the Earth has moved on its orbit, the line will be parallel to the previous day's line and will miss the Sun. It will take a little longer for the line between the center of the Earth and that spot on the Earth's surface to once again be pointed at the center of the Sun.
Thus a solar day, the time between a point on the Earth having the same position relative to the Sun, is a littler longer than a sideral day, being 24 hours or 86,400 seconds.
A difference of 235.9 seconds or 9.829 minutes isn't much of a difference between Earth's sidereal and solar days, but on some planets there could be a big difference that could be important in designing a solar system.
I note that when Dole wrote that a planet's day had to be more than 2 or 3 Earth hours long he meant the sidereal day, and when he wrote that a habitable planet's day should be less than 96 hours or 4 Earth days long he meant the solar day of the planet.
Part Six: Orbits.
I thnk that it would be necessary for th eorbital period of the planet around one star and the orbital period of the stars around their center of gravity to have the same length, and thus the angles change at the same rate, for the two stars to constantly have the same relative position in the sky of the planet.
Suppose that your planet orbits a G2V star with the mass of the Sun and the Luminosity of the Sun at a distanc eof 1 AU and thus has an orbital period of 1 Earth year. Suppose that the other star is also a G2V star with 1 solar mass and 1 solar luminosity and there is at least 5 AU between the two stars.
According to this planetary orbital calculater if a "planet" (in this case one of the two stars) with the mass of the Sun orbits the other tar with the mass of the Sun at a distance of 5 AU the orbital period will be 7.90453 Earth years.
https://calctool.org/CALC/phys/astronomy/planet_orbit
So the orbital period of the two stars would be 7.9 times as long as the orbital period of the planet around one of the two stars, so that would not work.
Suppose that your planet orbits a K2V star wiht 0.82 the mass and 0.37 the luminosity of the Sun and the K2V star more or less orbits a F2V star with 1.46 the mass and 5.13 times the luminosity of the Sun, at a distance 5 times the separation of the planet and the K2V star.
If the planet orbits the K2V star at a distance where it gets as much radiation from that star as Earth gets from the Sun - I call that the Earth equivalent Distance or EED - it will have about the same temperature range as Earth. Since the K2V star would have 0.37 the the luminosity of the Sun, and 0.60827 is the square root of 0.37, the EED of the K2V star would be about 0.608 AU.
If the F2V star was 5 x 0.608 AU from the K2V star it would be 3.01435 AU away.
Putting the mass of the planet as 1 Earth mass, the distance as 0.60827 AU, and the mass of the star as 0.82 solar mass into the planetary orbit calculater it give the orbital period of the planet around the K2V star as 0.523797 Earth. Putting the mass of the K2V star as 0.82 solar mass, the orbital distance as 3.0 AU, and the mass of the F2V star as 1.46 solar mass into the calculator, it gives an orbital period of the K2V star around the F2V star of 4.30116 Earth years. That is 8.2115017 times as long as the orbit of the planet around the K2V star.
So let's make the larger and more distant star an A2V star with 1.98 times the mass and 23.99 times the luminosity of the Sun. The orbital period of the planet around the K2V star will remain 0.523797 Earth years. And according to the calculator the orbital period of the K2V star around the A2V star at a distance of 3.01435 AU will be 3.71864 Earth years, or 7.0993915 times as long as the orbital period of the planet around the K2V star.
And I guess if you made the nearer star less massive and the farther star more massive, you would eventually get a combination of masses, luminosities, and distances which would make the orbital period of the planet around the near star and the orbital period of the nearer star around the farther star equal in length, so the two stars might possibly constantly remain in the same positions relative to the planet.
But you would run into the problem that less massive and less luminous stars would start to tidally lock planets in their habitable zones. Thus the smaller stars would always appear in the same position from a spot on the side that faced them, them and would never appear to rise or set, and would not be seen at all from the other side of the planet. That problem could be solved by giving the planet a very large moon, or make the planet a double planet, or make the planet actually a planet sized moon of a giant planet. In that case the planet would tidally locked to its companion world and not to its star and the star would appear to rise and set on the "planet". And that could enaable a possibly habitable planet to orbit a much smaller star than otherwise.
Andother problem would be that as the nearer star got dimmer and the farther star got brighter, eventually the planet would get as light and heat from the farther star as from the nearer star, and if the trend contnued the planet would get more radiation from the farther star than from the nearer star.
And as the more massive star gets more massive and more luminous, its lifetime on the main sequence of steller development before it becomes a red giant will get shorter and shorter. The F2V star, for example, would only have a main sequence period of about three billon years, probably not long enough for its palnets to naturally develope oxygen rich atmospheres, and inreasingly massive stars would last for shorter and shorter periods.
A writer could psssibly get around that by having an advanced society terraform a planet in the system and inhabit it for one million years, and then abandon it as the red giant phase of the larger star approached, and sell the planet to another society who might inhabit it for ten thousand years and then abandon it as the red giant phase got nearer, and sell it to your romantic protagonist with a habitabiity guarantee for only one hundred Earth years, which the protagonist might consider long enough.
Part Seven: A more complicated star system.
I imagine a more complicated star system where a planet, perhaps artificially terraformed for habitability, orbits around a black hole, and probably is tidally locked to the black hole, which would not emit any radiation or be seen in the sky of the planet. And there are two stars obiting the black hole at the same distance, but 180 degrees apart, and those two stars provide light and heat to the planet. Being 180 degrees apart i ntheir shared orbit, one star would rise while the other star was setting as seen from the planet.
But unfortunately that type of orbit has been imagined before, in the situation where a planet described as a "Counter-Earth" orbits on the oppsite of the Sun from the Earth, and so is always hidden by the Sun from detection on the surface of Earth.
Furthermore, a Counter-Earth would eventually be visible from Earth because the gravitational forces of the other planets on it would make its own orbit unstable. Venus has 82% of the mass of Earth and would come within 0.3 AU of the location of a Counter-Earth every 20 months, providing considerable gravitational pull that over the years would move its orbit into sight of observers on Earth.[18] If a Counter-Earth was much smaller than Earth, its location at the "Sun–Earth L3" Lagrangian point (see diagram) would mean the combined gravitational pull of the two large masses of Earth and the Sun would provide "precisely the centripetal force required to orbit with them". But a small planet would be influenced more by the orbit of Venus, Mars and Jupiter, making it even more unstable.
https://en.wikipedia.org/wiki/Counter-Earth#Scientific_analysis
So if you imagine the invisible black hole in the place of the Sun, the two stars in the places of the Earth and the Counter-Earth, and the planet in the place of another planet in our solar system, eventually that planet's gravity would perturb the orbits of the two stars so that they were no longer exactly 180 degrees apart in the sky of the planet. The sometimes one star wuld be visible, sometimes two stars would be visible, and sometimes no stars would be visible from the planet.
So calculations would have to be made to see if the two stars could remain 180 degees apart for long enough for the purposes of the story.
And I have an idea to improve the situation which I will describe in a later continuation of this answer.
Part Eight: Rings of stars or stellar mass objects.
The blog PlanetPlanet by Astrophysicist sEan Raymond has a section called the Ultimate Solar System about designing imaginary star systems with as many habitable planets as is physically possible.
https://planetplanet.net/the-ultimate-solar-system/
In this post https://planetplanet.net/2017/05/03/the-ultimate-engineered-solar-system/ he mentions a paper by Smith and Lissaeur discussing planets sharing the same orbit around a star (which can also apply to stars sharing the same orbit around a much more massive object like a very massive black hole).
The calcuations of Smith and Lissaeur showed that a number of equally massed objects equally spaced could share a single orbit around a larger object, and their orbits would be long term stable. Their calculations cover systems with between 7 and 42 objects in the orbiting ring.
https://ui.adsabs.harvard.edu/abs/2010CeMDA.107..487S/abstract
So naturally Rayond was inspired to design a solar system with several rings of habitable planets orbiting their star. With seven orbiting planets they would be spread out 51.428571 degrees apart along the orbit. With 42 planets per orbit they would be seperated by 8.5714285 degreees.
As Raymond says:
I can only think of one way our 416-planet system could form. It must have been purposely engineered by a super-intelligent advanced civilization. I’m calling it the Ultimate Engineered Solar System.
And in https://planetplanet.net/2018/05/30/the-black-hole-ultimate-solar-system/ Raymond designs solar systems with rings of planets orbiting a giant black hole, the planets being illuminated and heated by one or more rings of stars also orbiting the black hole.
And like the previous example, such a system would have to have been constructed by an advanced civilization instead of forming naturally in such a convenient way.
So you can imagine a system with a giant black hole, with the mass or hundreds or thousands of stars at least, and your habitable planet orbiting close to the black hole, and farther out from the black holethere is a ring of stellar mass objects sharing the same orbit. But only two of those stellar mass objects are actually luminous star illuminating the planet, and those two stars are on opposite sides of the ring 180 degrees apart, and so when one star is rising the other star will be setting.
If you try making a diagram of the system you will see that the more times the radius of the Stars' orbit around the black hole exceeds the radius of the planet's around the black hole, the less difference the movement of the palnet around the black hole will make in the apparent directions to the two stars as seen from that moving planet. So you will wantto make the two stars orbit the black hole at many times the orbit of the planet around the black hole.
This may be a complicating factor. The planet will be tidally locked so one side always faces the black hole and the other side never faces the black hole. So the planet will rotate on its axis once in every orbit around the black hole. But the planet will have to orbit the black hole far enough to beoutside the roche limit of the black hole, and also within the distance range where it has a day of the proper length. If the planet turns on its axis too fast it will fly apart and never be habitable. So a planet tidally locked to the black hole would have a rotation period equal to its orbit around the black hole, which thus has to be long enough for the planet not to spin apart.
And if the planet is orbiting the black hole far enough to have a slow enough spin, and if the orbit where the stars orbit is several times as far from the black hole as the planet's orbit, the stars might not be close enough to keep the planet warm enough for life.
Another problem is that the number of stellar mass objects in the shared orbit has to be even in order for the two stars in that orbit to be 180 degeess apart on opposite sides of the orbit. So the number of stellar mass objects would have to be 8, separated by 45 degrees, or 10 separated by 36 degrees, or 12 separated by 30 degreees or 14 divided by 25.714285 degrees, and so on.
I see that 180 degrees can be evenly divided by 45, 36, 30, or 25.714285 degrees, so presumably any even number of objects up to 42 in the orbit would enable the two stars to be 180 degreees apart.
So now the question is, what would the stellar mass objects be, to share an orbit with two stars of the same mass as them without also being stars?
Part Nine: Stellar mass objects in the ring.
So one way to get a bunch of stella rmass objects in the ring but make only two of them stars, is to make each object actually a binary of less than stellar mass objects.
To recap, I earlier suggested that a habitable planet would probably have a mass between 0.25 and2.0 Earth mass.
The Sun has about 330,000 times the mass of Earth.
The planetjupiter has 317.8 times the mass of Earth.
And theoretically the dividing line between the most massive planets and the least massive brown dwarfs should be about 13 Jupiter masses or 4,131.4 Earth masses or 0.0125193 Solar mass.
The dividing line between the most massive brown dwarfs and the least massive stars should be about 75 times the mass of Jupiter or 23,835 Earth masses or 0.072222 Solar mass to about 87 times the mass of Jupiter or 27,648.6 Earth masses or 0.0837836 Solar mass.
So imagine a ring of objects each with a mass of 140 Jupiter masses or 44,492 Earth masses or 0.1348242 solar mass around the black hole. Except that only two of them are actually single masses, stars with a mass of 0.1348242 Solar Mass. The other objects in the orbital ring can each be binaries, each having two brown dwarfs each with a mass of 70 Jupiter masses or 22,246 Earth mass or 0.674121 Solar mass. The brown dwarts would look like stars or planets in the sky of the planet orbiting the black hole.
Stars with only 0.1348242 Solar mass would be very dim red dwarf stars. They would be more massive and luminoous than spectral class M6V stars with 0.102 Solar mass and 0.001 Solar luminosity, and less massive and luminous than spectral class M5V stars with 0.162 Solar mass and 0.003 Solar luminosity.
https://en.wikipedia.org/wiki/Red_dwarf
And there might be problems with having the planet orbit far enough from the giant black hole while the dim red dwarfs orbit far enough beyond theplanet's orbit to always seem to be separated by 180 degrees while still being close enough to the planet to keep it habitabley warm.
Part Ten: White dwarfs in the ring.
Or the ring can have more massive stars, including two main sequence stars and a bunch of white dwarfs stars in the other positions.
White dwarf stars were once main sequence stars which have lost much of their original mass and are now very small and dense and emittonly a tiny fraction of the light that main sequence stars of the same mass emit.
Calculations indicate that all white dwarfs must have between about 0.5 and about 1.4 times the mass of the Sun.
A M1V class star would have 0.5 the mass and 0.041 the luminosity of the Sun. A main sequence star with 1.4 times the mass of the Sun would be
between a F4V with 1.38 the mass and 4.17 the luminosity of the Sun and a F3V with 1.44 the mass and 4.68 the luminosity of the Sun.
The white dwarf stars in the system would appear as very bright stars in the night and even the day sky of the planet.
Part Eleven: A ring of neutron stars.
Possibly you might find the white dwarfs stars in the ring too bright for your planet.
A neutron star is the remnant of a once much more massive star. Neutron stars, much denser and smaller than white dwarfs, have masses in the range of about 1.1 to 2.16 Solar mass.
A star with 1.1 times the mass of the sun would be a little less than an FOV class star with 1.13 the mass and 1.66 the luminosity of the Sun.
https://en.wikipedia.org/wiki/F-type_main-sequence_star
A star with 2.16 the mass of the Sun would be a little less than an A0V class star with 2.18 the mass and 38.02 the luminosity of the Sun.
https://en.wikipedia.org/wiki/A-type_main-sequence_star
The luminosity of neutron stars varies a lot. Most detected neutron stars are pulsars emitting pulses of radio waves.
Pulsar planets receive little visible light, but massive amounts of ionizing radiation and high-energy stellar wind, which makes them rather hostile environments.
https://en.wikipedia.org/wiki/Neutron_star#Planets
So Neutron stars might be too dangerous for life on your planet.
Part Twelve: Black Holes.
Maybe the intermediate mass or supermassive black hole that your planet orbits closely has a ring of stellar mass black holes orbiting farther out. Two of the objects can be stars 180 degreees apart with the same mass as each black hole.
The least massive known stellar mass black hole has about 5 times the mass of the Sun. The most massive stellar black holes should have about 45 to 60 times the mass of the Sun.
A B4V class star would have 5.10 the mass and 776 the luminosity of the Sun.
https://en.wikipedia.org/wiki/B-type_main-sequence_star
And a star with 45 to 60 times the mass of the Sun would be a class B star thousands of times as luminous as the Sun.
So while the black holes in the ring would not be visible, stars with the same mass would be very luminous and short lived.
So you would probably be better off using a ring of six white dwarfs and 2 main sequence stars with the same mass as the white dwarfs.
