I'm currently creating my own binary star system which includes a circumbinary planet that has two moons. For simplicity's sake, the stars, planet, and moons are comparable in size to our own Sun, Earth, and Moon. The planetary system is approximately 1 AU from the center of mass of the two stars. What is the likelihood of binary solar eclipses from the perspective of both the planet and its moons?
The easiest way to implement a "binary solar eclipse" is to have your planet actually be a moon orbiting a gas giant - every orbit, the gas giant will block both stars for an extended period. But that's probably not what you're after.
(As a side note, putting the planet, suns, and moons at the same sort of sizes and distances as our solar system is just asking for trouble. The easiest fix is probably just to make the two suns smaller than ours, which also gives you more of a margin for eclipses. I'm going to assume that's what you're doing for the rest of this question.)
The best I can say about a binary solar eclipse - that is, an event where both suns of a binary star system are eclipsed by the moon (or moons) of a planet orbiting that star system - is that the laws of physics do not explicitly prohibit it. Like the other guy suggested, the easiest way is probably to use one large moon. However, if you want two moons eclipsing two suns, then a much more interesting possibility opens up.
The bad news is that there is basically no way for such a system to arise naturally. Even just getting an earth-sized planet with two moon-sized (or half-moon-sized) moons is spectacularly improbable - the going theory on how we probably got our moon involves a catastrophic planetary collision in the early solar disc. That collision happening under exactly the right circumstances that it did - leaving a planet + moon instead of another asteroid belt - is so unlikely that it's been floated as an answer to the Fermi Paradox. We haven't been contacted by aliens because you need a moon like ours to develop intelligent life, and we're the only planet that got that lucky.
When writing realistic fiction, it's easy to overlook that it's still fiction. All the physics we're discussing here was developed to explain what we see in the real world. And when we're explaining the real world, we have to fit it into natural systems, because that's the only kind available.
But if the system couldn't have arisen naturally, it must have arisen unnaturally - and since you're writing fiction, you can do that. You don't have to hew to strict natural processes. You can make your solar system artificial.
The most reliable way to do this (actually, I think it's the only way I've ever seen) is to say that the system was engineered by some ancient bunch of aliens who don't visit anymore. Maybe they are / inspired the gods that the local civilizations now worship, maybe they're just a big question mark. But "aliens mucked about with my solar system a zillion years ago" justifies a lot of phlebotinum and other fun plot devices.
Want magical artifacts? Leftover alien technology plus Clarke's third law. Want dragons and/or other mythical creatures? Alien bio-engineering. Want to spice up the "vanished ancient alien precursor" trope? It was actually the equivalent of a grad school project, and the reason the aliens aren't around anymore is that they forgot about you and are doing something else now.
This approach also solves a lot of the other problems with binary star systems (like stability), by way of "the aliens were better at orbital mechanics than we are and figured out how to make problems x, y, z not happen."
If by "binary solar eclipse" you were hoping for two suns in the sky that are simultaneously eclipsed by two moons then you're probably going to be disappointed.
Thing is, the two suns are going to be approximately the same size, whereas the two moons are going to have to be quite different sizes, or they'll end up gravitationally interfering with each other and one of the two is going to get ejected from the planetary orbit.
What you can have is a simultaneous conjunction of the two suns and a single large moon. The frequency of such an event very much depends on the details of your planetary system, especially orbital radii, inclinations and diameters of the various bodies that are involved, so it isn't practical to give you a hint as to their likelihood.
For simplicity's sake, the stars, planet, and moons are comparable in size to our own Sun, Earth, and Moon. The planetary system is approximately 1 AU from the center of mass of the two stars
Your Earth would be roasted by being subjected to two sun's worth of heat and light. I also believe that your stars would have to be much too close together... I'm not sure what the minimum separation of a binary is offhand, but I am aware that there is a critical radius below which a circumbinary orbit is unstable, and I think your planet is likely to fall within it.
If you wanted to make everything as hard-scifi-plausible as possible, you could ask a separate question about the configuration of your system... there are people more familiar with that sort of thing that me, though I could probably give it a stab.
Alternatively, you could just take the soft scifi approach and handwave in your desired setting because it would look awesome, and honestly that's good enough.
So the star is binary and the moons are binary, the planet having two moons?
So a "binary solar eclipse" could be:
The two suns are eclipsed one after another by one of the moons.
One sun is eclipsed by one of the moons, and then by the other moon.
the two suns are eclipsed at the same time by the different moons. Moon A eclipses sun A while moon B eclipses sun B.
All three versions would require that three planes be identical or only slightly tilted.
Those planes would be:
the plane in which the two stars orbit their center of mass.
the plane in which the planet orbits the two stars.
the plane in which the two moons orbit the planet. And of course the two moons might orbit in two separate planes. In that case they would have to be only slightly tilted relative to each other and the other orbital planes.
I note that if the two stars are both equal in size, mass, and luminosity to the Sun, and the planet orbits at the same distance from them as Earth does from the Sun, the planet will beceive twice as much radiation from them as Earth gets from the Sun, and it will be considerabley hotter than Earth, perhaps hot enough to be lifeless.
The square root of two is 1.4142135. If the stars are both as luminous as the Sun, your planet would have to orbit at 1.414 times Earth's distance from the Sun, and thus at 1.414 Astronomical Units from the center of gravity of the two stars.
Because the combined mass of the two stars, would be twice that of the Sun, and because the planet would orbit at 1.4142135 AU, the length of the planet's year or orbital period around the two stars, would be different from that of Earth.
According to this orbital period calculator, http://www.calctool.org/CALC/phys/astronomy/planet_orbit
the orbital period or year of the planet would be: 1.8900 Earth years.
Or if you want to keep your planet at a distance of 1 AU from the center of gravity of the two stars, you should make their total luminosity the same as the Sun's. So the luminosity of star A plus the luminosity of star B should equal the Sun's luminosity.
If the two stars were of equal luminosity, they would be on the border between typical spectral class G9V stars with 0.9 the mass of the Sun, 0.853 the radius of the Sun, and 0.55 the luminosity of the Sun one one hand, and typical spectral class K0V stars with 0.88 the mass of the Sun, 0.813 times the radius of the Sun, and 0.46 times the luminosity of the Sun, on the other hand.
Note that at a distance of 1 AU the two stars would have apparent diameters only about 0.81 to 0.86 that of the Sun, so moons with a slightly smaller angular diameter as seen from the planet than the Moon has as seen from Earth, would suffice to cover the two stars in an eclipse, although larger angular diameters would also do.
Since the total mass of the two stars would be about 1.8 times the mass of the Sun, the planet at a distance of 1 AU would need a faster orbital speed, and thus the length of its year would be significantly shorter than an Earth year.
The orbital period calculator gives 0.745228 Earth years.
Of course the two stars do not have to be twin stars, they can have significantly different masses, sizes, and luminosities and still have their total luminosity equal to that of the Sun.
The two stars do not have to have a total luminosity equal to that of the Sun, and the planet doesn't have to orbit at any specific distance from the center of gravity of the two stars, so long as the planet's orbit is within the combined circumstellar habitable zone of the two stars, and so long thas those two stars are both within the range of star types considered capable of having habitable planets (unless of course the planet in the story is supposed to be uninhabitable for humans or for any other type of life).
If you want the temperatures to be very similar to those on Earth, you should probably put the planet's orbit in what I call the Earth Equivalent Distance, or EED, the distance at which it will receive the same amount of heat and light from its star(s) as Earth gets from the Sun.
Because the planet will need to orbit the center of mass of the two stars at a distance several times their separation, it will orbit the center of pass much slower than the stars orbit it. Thus it will be impossible for the two stars to always show the same angle as seen from the planet. Instead observers on the panet will see the stars orbiting each other, with their angular separation ranging from zero degrees when one star is in front of the other, out to the maximim possible visual separation.
It is posssible for the two stars to orbit very closely together, with separation of only a few million miles, and even a few hundred thosuand miles. They could also orbit with a separation far too large for their habitable zones to be combined, orverlap, or even touch each other. Since you want your planet to orbit both of the stars in the combined habitable zone, you will want the separation between the stars to be fraction of the radius of the planet's orbit.
The smallest known ratio of the semi-major axis of the orbit of an exoplanet in a circumbinary orbirt around two stars to the orbital separation of the two stars it orbits is siad to be Kepler 16 b. Kepler 16 A & B are separated by about 0.22 AU and the planet Kepler 16 orbits their center of gravity with a semi-major axis of about 0.7048 AU, about 3.14 times the separation between the stars.
And it is considered very improbable for a planet to have a long term stable circumbinary orbit if the ratio of the planet's orbit to the separation of the two stars is much smaller than that. In such a situation the different pulls of the gravity of the two stars would tend to destablize the planet's orbit.
Assume that the semi-major axis of the planet's orbit can be as little as 3 times the separation of the two stars, or three separation units (su). When the planets and stars are positioned so that the planets sees the full separation between the two stars, they will be separated by 1 su and be at a distance of 3 su. So as seen from the surface of the planet the two stars would then be separated by an angular separation of about 19 degrees of arc.
And most circumbinary planets would be orbiting with much higher ratios than such a minimum possible ratio between the orbit and the separation of the planet. And so the maximum possible angular separation between the two stars as seen from their circumbinary planets should be much less than 19 degrees of arc.
Suppose that your planet has two large moons which orbit the planet at different distances. In that case they will periodically be lined up so that the nearer one passes in front of the outer one and so eclipses or occults it.
Since it is impossible for the planet to orbit the center of gravity with the same orbital period as the orbital periods of the two stars around the center of gravity, it is impossible for the two stars to always have the same angular separation as seen from the planet. So the separation between the two stars will get larger and smaller over time. Sometimes they will appear at their maximum possible separation, and sometimes they will appear at smaller separations, down to zero separation when one star passes in front of the other.
There such be a specific regular period between the eclipse of star A by star B, and the following eclipse of star B by star A.
And there will also be specific regular period between the times when a moon, say the outer moon, happens to be lined up wbetween the planet and the center of gravity of the two stars.
And there will be a specific regular period between successive times when the inner moon passes in front of the outer moon and eclipses or occults it.
And there will be a theoretical period of time which can be evenly divided by each of those three previous periods of time. The length of that period of time will vary greatly depending on the various ratios between the other periods.
And at the beginning and end of each such period of time, the two stars and their center of mass will be lined up with the planet, so the nearer star eclipses the farther star as seen from the planet, and at the same time the outer moon passes in front of the two stars and eclipses both of them, and at the same time the inner moon passes in front of the outer moon and the two stars and eclipses both stars and the outer moon.
It is perfectly possible for the interval between two successive triple eclipses of the two stars and the outer moon to be so long that it will never happen even once in all the billions of years the two stars will remain on the main sequence betore one of them swells up into a red giant and swallows the other star and perhaps the planet and its two moons as well.
So possibly when such an event is calculated to happen soon, it might be demonstarted that it will be the first and only time it will ever happen in the entire existence of the planet, and people will be very eager to see such a once in a (planetary) lifetime spectacle.
And on the other hand, if the orbital periods of the various bodies, and thus the periods between successive linings up of two bodies with the planet, are chosen correctly, the multiple eclipse could happen as often as once a millennium, once a century, once a decade, once a year, or even more often.
If such a multiple eclipse or occultation can happen, events when one of the moons eclipses both of the stars will be much more common, and events where both of the moons eclipse one of the stars will also be much more common. Events where one of the moons eclipses one of the stars should be more moon still. The most common line ups of all will be when both of the stars are lined up as seen from the planet without being eclipsed by either of the moons, and when both of the moons are lined up without eclipsing either of the stars. And almost all of the time there won't be any lineups or eclipses at all.
And I think I will enter a second answer discussing the situation where the two moons of the planet orbit at the same distance from the planet.
If the binary star system is an eclipsing binary then the eclipes would be frequent but the frequency of such eclipses would depend on how far away both stars are from each other. If they are close( less than 1 AU away from each other then an eclipse should happen at lest once in every month).
If both stars are more than 1AU of distance away from each other then an eclipse would happen just once every a few months or just once in every few years.