This is a really long answer which elaborates on the answer by BIOStheZerg, pointing out some of the factors to be considered when designing a star system according the suggested answer of BIOStheZerg. I couldn't think of any other way to provide what AvengingEarth wants, so I thought that would fill in someof the details of BIOStheZerg's answer.
BIOStheZerg suggested that the planet should orbit in an S-type orbit, and thus orbit only one of the two stars, and the planet should be tidally locked to the star that it orbits, so that one side of the planet would always face the star and the other side would always face away from the star.
BIOStheZerg suggested that only the twilight zone of the planet would be habitable for the natives, and thus that the star would always be near the horizon of the planet from the twilight zone inhabited by the natives.
`A small change in the mass of a star will produce a much larger change in the luminosity of the star. Thus a planet will have orbit a less massive and dimmer star much deeper into the gravity well of that star, and the tidal braking of that star will slow down the planetary rotation much faster.
Thus the only tidally locked planets will be planets of low mass and very dim stars.
Astronomers for many years ruled out red dwarfs as potential abodes for life. Their small size (from 0.08 to 0.45 solar masses) means that their nuclear reactions proceed exceptionally slowly, and they emit very little light (from 3% of that produced by the Sun to as little as 0.01%). Any planet in orbit around a red dwarf would have to huddle very close to its parent star to attain Earth-like surface temperatures; from 0.3 AU (just inside the orbit of Mercury) for a star like Lacaille 8760, to as little as 0.032 AU for a star like Proxima Centauri (such a world would have a year lasting just 6.3 days). At those distances, the star's gravity would cause tidal locking. One side of the planet would eternally face the star, while the other would always face away from it. The only ways in which potential life could avoid either an inferno or a deep freeze would be if the planet had an atmosphere thick enough to transfer the star's heat from the day side to the night side, or if there was a gas giant in the habitable zone, with a habitable moon, which would be locked to the planet instead of the star, allowing a more even distribution of radiation over the planet. It was long assumed that such a thick atmosphere would prevent sunlight from reaching the surface in the first place, preventing photosynthesis.
This pessimism has been tempered by research. Studies by Robert Haberle and Manoj Joshi of NASA's Ames Research Center in California have shown that a planet's atmosphere (assuming it included greenhouse gases CO2 and H2O) need only be 100 millibars (0.10 atm), for the star's heat to be effectively carried to the night side. This is well within the levels required for photosynthesis, though water would still remain frozen on the dark side in some of their models. Martin Heath of Greenwich Community College, has shown that seawater, too, could be effectively circulated without freezing solid if the ocean basins were deep enough to allow free flow beneath the night side's ice cap. Further research—including a consideration of the amount of photosynthetically active radiation—suggested that tidally locked planets in red dwarf systems might at least be habitable for higher plants.
So on one extreme, a tidally locked planet might have so little heat circulation that all of the water and air might freeze out on the permanent night side, and there might not be any life on the planet.
And on the other extreme, there might be enough air and water circulation in a tidally locked planet that temperatures will be fairly good in all parts of the planet, and life might not be restricted to a narrow twilight zone.
So a situation where the natives are restricted to a rather narrow twilight zone where the star always appears low on the horizon, the desired situation, might be a rarity between dead tidally locked planets on one hand and tidally locked planets with decent temperatures in every part on the other hand, where the natives would not be restricted to a twilight zone and might be able to see the star high above the horizon in some places.
But since it has not been proved to be totally impossible yet we can accept that the planet might have such a situation.
As said earlier, a tidally locked planet would have its rotation rate slowed to match its orbital period around the star, so that one side always faced the star and the other side always faced away from the star, and the star would always be low on the horizon at the twilight zone. And it would have to be orbiting very close to a dime low mass star.
Assuming that the natives of the planet have environmental requirements similar to humans, the discussion of tidally locked planets in Habitable Planets for Man, Septen H. Dole, 1964. 2007, may be useful. Dole didn't believe that tidally locked planets could be habitable, so he ruled out any stars so dim that planets in their habitable zone would be tidally locked.
Dole discusses the masses of stars capable of having habitable planets on pages 67 to 72.
Because more massive stars with larger habitable zones consume their nuclear "fuel" much faster than less massive stars, they spend less time in their main sequence stages before becoming red giants, and then white dwarf stars (or possibly neutron stars or even black holes). Since a planet should take billions of years to become habitable for humans, a star that is too massive will not spend enough time on the main sequence with a steady luminosity for its planets to become habitable for humans.
The only stars that conform with the requirement of stability for at least 3 billion years are main-sequence stars havng a mass less than about 1.4 solar masses - spectral type F2 and smaller - although the relationship between mass and time in the main sequence is probably not know with great accuracy and is subject to future revisions (see Figure 25).
Dole, believing that tidally locked planets would be uninhabitable for humans, unlike the assumptions made in the question, then discussed the stellar masses which would result in tidal locking of planets in their habitable zones, which Dole calls "ecospheres".
...habitable planets can exist in ecospheres only around stars having masses larger than about 0.72 solar mass. A "full" ecosphere can exist around primaries of stellar mass greater than about 0.88 solar mass, but the ecosphere is narrowed by the tidal braking effect for primaries of lesser mass until it disappers when the stellar mass reaches about 0.72 solar mass. The range in mass of stars which could have habitable planets is thus 0.72 ot 1.43 solar masses, corresponding to main-sequence stars of spectral types F2 through K. There is an extension of this range down to the larger class M stars (mass greater than 0.35 solar mass) for a special class of planets with large satellites. This will be discussed in the next section.
So Dole believed that for stars between 0.72 and 0.88 solar mass, part of the "ecosphere" or circumstellar habitable zone would be close enough to the star that planets orbiting there would e tidally locked to the star, while for stars lower than 0.72 stellar mass the entire "ecosphere" or circumstellar habitable zone woudl be close enough to the star that palnets within it would be tidally locked.
Since Dole believed that all tidally locked planets would be uninhabitable for humans, he disregard them and the dim stars they orbited.
But the question, and the answer of BIOStheZerg, require that the planet be both habitable and tidally locked, so that one star in the system always appears low on the horizon in the twilight zone of the planet. Since that has not been proved to be impossible, that is the set up I am using in my elaboration of BIOStheZerg's answer.
Astronomes have already discovered a number of planets orbiting within the habitable zones of their stars, and some orbit dim stars close enough they should be tidally locked.
Wikipedia has a list of potentially habitable exoplanets orbiting within the habitable zones of their planets. The ones orbiting K1 class stars and less massive stars should be tidally locked to their stars.
The ones orbiting spectral type K and M stars include planets with orbital periods of hunderds of Earth days, including 289 days, 267 days, 259 days, 247 days, 198 days, 197 days, 177 days, 168 days, 147 days, 129 days, 122 days, 112 days, and 101 days.
Other exoplanets in the list, most of them, have orbital periods between 10 and 99 Earth days.
And some of them have orbital periods even shorter.
TRAPPIST-1f 9.2 days, TRAPPIST-1e 6.1 days, Teegarden b 4.91 days, and TRAPPIST-1d 4.05 days.
So these examples prove that a tidally locked planet in the habitable zone of its star could have an orbital period as short as 4.05 Earth days, and quite possibly as short as 4.000 Earth days. Considering how dim TRAPPIST-1 and Teegarden's star are, I don't feel it is safe to imagine a habitable planet having an orbital period much less than 4.00 Earth days long. Such tidally locked planets can also have orbital periods as long as 289 Earth days and probably longer.
The other star in the system.
To design a working star system, for AvengingEarth who asked theoriginal question, it will be necessary to decide whether the other star will provide both significant heat and light to the planet, or merely significant light.
The human eyes as a vast ability to adapt to a wide range of ilumination.
Humans can see fairly well and function in a moonlit night. The dimmest moonlight, from a new moon, is magnitude -2.5, while the brightest moonlight, from a full moon, is magnitude -12.9, about 10,000 times brighter.
The Sun's apparant magnitude in broad daylight has an apparent magnitude of -23.0, which is about 400,000 times as bright as the full moon.
So it would be easy to make the other star in the system close emough to be many times as bright as the full moon on Earth, and thus make it easy for the native sto see whenit is above the horizon, while still much less bright than the Sun as seen from Earth.
A light source which is twice as fr away as an equally bright light source will appear a quarter as bright. A light source which is 3 times as far away will a ninth as bright. A light source 4 times as far away will appear 1/16 as bright, one 5 tiems as far away will appear 1/25 as bright, one 6 times as far away will appear /36 as bright, one 7 times as far awa will appear 1/49 as bright, one 8 times as far away will appear 1/64 as bright, one 9 times as far awy will appear 1/81 as bright, and one 10 times as far away will appear 1/100 as bright.
And if the light source is moved to 100 times its original distance, it will appear 1/10,000 as at the original distance.
So assuming that the two stars have equal luminosity, and the nearer star gives the planet the same amount of light as the Earth does, the farher star could be 100 times as far from the planet as the nearer star and still give the planet 1/10,000 as much light as the nearer star, which would be about 40 times the brightness of a full moon on Earth, and certainly enough light to see where someone is going.
So there would be no problem with designing a star system where the farther star gives the planet much less light than the nearer star does, but still enough light to see well and see where you are going. Certainly enough light for the natives of the planet to base their calendar on the rising and setting of the farther star.
But designing a star system where the farther star gives enough radiation to the planet to have a significant effect on the planet's temperature is a different story.
The planet Saturn is about 10 times as far from the Sun as Earth is, so the sunlight at Saturn is 1/100 times as bright as on Earth, which is still about 4,000 times as bright as the light of the full moon on Earth. You could see well enough in sunlight to walk around in your spacesuit on a moon of Saturn without using artifical light, and you would definitely notice how dark it got when the Sun set on that moon.
But the distance from the Sun makes a much more important difference in temperature. The average surface temperature on Titan, the big moon of Saturn, is 97 degrees K, or minus 179.5 degrees C, or minus 291.1 F.
Getting the farther star in the system to be luminous enough, and/or close enough to the planet, to contribute a signficant degree of heat to the planet will be much more difficult.
If a planet orbits a star in an S-type orbit in a binary star system, the other star sould be several tiems as far from the planet as the near star for the orbits to be stable over long periods of time.
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.
So if the farther star's orbit periodically takes it less than five times as far from the nearer star as the planet orbits, the orbital stability is "not guaranteed".
If the two stars in the system ahave the same luminosity, the farther star can contribute no more than 1/25, or 4 percent, of the heat of the planet, and possibly a lot less.
The farther star would have to be at least 25 times as luminous as the nearer star in order for it to contribute equally more more to the temperature of the planet, and that is if the farther star is at is minimum possible distance.
Can a system with a habitable planet have one star that is at least 25 times as luminous as the other star? As quoted above, Dole said that a star had to have less than about 1.43 solar masses and be a spectral type F2 or less to have a habitable planet.
Alpha Corvi is spectral type F1 or F2, has a mass of 1.39 solar mass, and 4.91 times the luminosityof the Sun. I suspect that the most luminous star that could be old eneough to have a habitable planet would be somewhere between 4.5 and 5.0 times the luminosity of the Sun.
TRAPPIST-1 is a spectral type M8 star with 0.08 the mass of the Sun and a luminosity of about 0.00055 that of the Sun. Thus it is probably almost as dim a star as couldp possibly have habitable though tidally locked planet. That makes a difference of about 8,180 to 10,000 between the brightest and the dimmest possible stars with habitable planets.
So if the farther star is as much as 10,000 times as luminous as the nearer star, and is as little as 5 times as from from the planet as the nearer star, it could give the planet as much as 400 times the heat and light as the nearer but much dimmer star does.
But in order for the planet not to be overheated and too hot for life, it would have to be within the habitable zone of the farther star and it would also have to be far outside the habitable zone of the nearer star. And I don't know if even the dimmest star could be so dim that a planet far beyond its habitable zone would still be tidally locked to it.
Thus the farther star in the system could be about 90 to 100 times as far from the planet as the nearer star, and still provide equal amount of heat to the planet if it is about 8,180 to 10,000 times as luminous as the nearer star.
The farther star could provide up to 10 percent of the heat of the planet if it was 316 times as far away as the nearer star, if it was 10,000 times as luminous as the nearer star.
The farther star could provide up to 1 percent of the heat of the planet if it was 1,000 times as far as the nearer star, if it was 10,000 times as luminous as the nearer star.
But it would be a very rare binary star system where both stars were within the mass range suitable for having habitable planets and one was as much as 10,000 times as luminous as the other.
Clearly, in many, possibly most, binary systems where there is a habitable planet in an S-type orbit around one of the stars, the farther star will be a very important source of light on the habitable planet, appearing much brighter than the full moon does on Earth, being bright enough that its periods will become units in the time keeping methods of any native people, and yet the farther star will be a totally insignficant source of heat for the planet.
And there is the problem that the nearer star is supposed to always be at the horizon where the natives of the planet live. That can only be true in the twilight zone of the planet. So the natives have to stay in that twilightzone all the time, or almost all the time. So they should have no reason to go to the light side of the planet, facing the nearer star, where the nearer star will rise higher in the sky, no longer being at the horizon.
And they should have no reason to go to the side of the planet facing away from the nearer star. If they do go there, the nearer star will not be visiple at all, instead of being on the horizon. So the farther star should not heat up the fouter side of the planet enough to make it comfortably warm for the natives. And if the farther star provides enough light to the outer side of the planet, it may cause plants to grow there, and animals to feed on those plants, and thus there may be reasons for the natives to go to the outer side.
So the light the planet gets from the farther star should be enough to be significant to the natives to see by and consider it important, but it should probably not be enough for photosynthesis by plants on the side of the planet away from the nearer star, and thus there should be little life on that outer side of the planet and little incentive for the natives to go there.
So what type of star should be the nearer star that the tidally locked habitable planet orbits in an S-type orbit?
In my opinion, the nearer star should be a very, very dim class M star, like TRAPPIST-1, a star so dim that the tidally locked habitable planet orbits it with almost the shortest possbile orbital period.
Depending on the mass and luminosity of a star dim enough to have tidally locked planets in its habitable zone, the orbital periods of those tidally locked planets should range from about 4.0 Earth days to 289 Earth days to maybe 300 Earth days.
If the other star, the farther star, in the system is a much more luminous, and thus more massive star, the orbits of the two stars around each other can be discussed as though the dim nearer star and its tidally locked planet orbit around the more luminous and more massive farther star.
Thus the farther star, nearer star, and tidally locked habitable planet system can be considered to be analogous to the Sun-Earth-Moon system, where the Moon orbits the Earth which orbits the Sun. The tidally locked habitable planet would be analogous to the Moon, the nearer star would be analogous to the Earth, and the farther star would be analogous to the Sun.
Since the Moon is tidally locked to the Earth, the Earth never appears to move (much) from its position in the sky of a spot on the near side of the moon. But since the Moon orbits completely around the Earth in a month, the Moon makes a full circle in relation to the distance stars in a month, and also makes a full circle in relation to the Sun in a month.
To be precise, the Moon makes a full circle with respect to the background stars in one sidereal month, equal to one complete orbital period around tbe Earth. That orbital period and sidereal month is 27.321661 Earth days. One sidereal month of 27.321661 Earth days is also the length of a sidereal day on the Moon, when the Moon rotates a full 360 degrees with respect to the distant stars in interstellar space. Thus the moon rotates 13.176358 degrees per day with respect to the distant stars.
But during a sidereal month or sidereal lunar day of 27.321661 Earth days, the Earth also travels along its orbit around the Sun. If the Earth's orbit around the Sun was perfectly circular, the Earth would travel exactly 0.9856 degrees of its orbit each day, and thus exactly 26.928477 degrees along its orbit during a sidereal month. That means that the moon would have to rotate another 26.928477 degrees to be aligned with the Sun the same way it was at the begining of the month, which would take another 2.04369 Earth days. But during those Earth days the Earth would travel another couple of degrees, so the Moon would have to catch up a little more, and so on.
The synodic lunar month is the time it takes for the Earth, the Sun, and the Moon to become aligned the same way they were at the start of a synodic lumar month, the time it takes for the phases of the Moon, as seen from Earth, and the phases of the Earth, as seen from the Moon, to go through a complete cycle and return to their original appearance. It is 29.530589 Earth days long.
And I think that the synodic month is also the length of the syndoic day on the Moon, the length of time between two successive sunrises or sunsets at a specific location on the Moon.
And if the tidally locked habitable planet is analogous to the Moon, the synodic day of the tidally locked planet will be the time it takes the farther star to appear to circle the sky once, the time between successive sunrises or sunsets at a spot on the planet. And that will be a period of time which the natives of the planet will nclude in their time systems and calendars.
Note that the sidereal day of the Moon is 27.321661 Earth days, and the synodic day of the Moon, the time between successive sunrises, is 29.530589 Earth days, a somewhat longer period.
And it seems to me that it would be impossible to make the synodic day of the tidally locked planet, the period which would be part of the calendars of the natives, as short as the sidereal day of the tidally locked planet, which would equal one orbit of the tidally locked planet around the nearer star.
I think that the longer the orbital period of the nearer star and the planet around the farther star is compared to the orbital period of the planet around the nearer star, the less difference there will be between the sidereal day and the synodic day of the planet.
And the shorter the orbital period of the nearer star and the planet around the farther star is compared to the orbital period of the planet around the nearer star, the more difference there will be between the sidereal day and the synodic day of the planet.
So the synodic day of the planet could vary between being slighly longer than the sidereal day of the planet to being much longer than the sidereal day of the planet.
Adn how long could the sidereal day of a tidally locked but habitable planet be?
Stars dim enough to tidlaly lock any planets in their habitable zones vary greatly in mass and luminosity and the lengths of the orbital periods of planets in their habitable zones.
The orbital periods of those tidally locked planets should range from about 4.0 Earth days to 289 Earth days to maybe 300 Earth days.
So if the orbital periods and thus sidereal days of tidally locked planets should range from about 4.0 Earth days to 289 Earth days to maybe 300 Earth days, and the synodic days of those tidally locked planets in respect to other stars in the system can vary between slightly longer than their sidereal days to several times as long as those sidereal days, the synodic days can be much longer than Earth years, as long as Earth seasons, as long as Earth months, as long as Earth weeks, or a few days long, depending on the length of orbital periods and sidereal days of those planets.
Thus I rather suspect that a really low mass and low luminosity star will be selected as the nearer star, so that the orbital period and sidereal day of the tidally locked planet will be as short as possible.