Could two planets trapped between two stars orbit each other? Say there are two stars, each exerting and equal force on a point in the midde of them. For simplicity, we'll say the two stars have the same mass and are equidistant from the point. Now suppose that there are two, equally massive planets, moving at a great speed around that point. Would the two planets be able to create a sustainable orbit between the two stars? Could such a situation ever naturally occur?
 A: Would you count this? Imagine two copies of the James Webb telescope in orbit. As you can see, the Webb telescope travels in a small circle as it orbits the Sun. Two could orbit in opposite sides of the circle.
It isn't quite what you asked, but almost. They orbit outside the Earth's orbit.
It also isn't really right to say they orbit each other. They are held by the gravity of the Sun and Earth. Their mutual attraction is very weak and isn't what makes them travel around each other.
But it fits the idea of what you asked. So yes, this is possible.

Edit
There is another possibility. It too isn't exactly what is asked for. But it requires equal size Suns, as Carl asks.
You start with two equal size Suns and equal size planets between them. But you make the planetary orbits so big that the Suns and planets form a square. This forms a Klemperer rosette. All 4 rotate at the same speed. It still isn't stable, but it is more stable than having planets in small orbits rotating with a different period from the Suns.
A: This wiki article says , yes

In astronomy, a double planet (also binary planet) is a binary system where both objects are planets, or planetary-mass objects, that share an orbital axis external to both planetary bodies.

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There is debate as to what criteria should be used to distinguish "double planet" from a "planet–moon system". The following are considerations.


At its 2006 General Assembly, the International Astronomical Union considered a proposal that Pluto and Charon be reclassified as a double planet,[2] but the proposal was abandoned in favor of the current IAU definition of planet.

Addition after comment
So there is no problem for two planets to revolve about each other and also their center of mass about a star.
So the doubt in your question is not about a binary system, but if there can be a stable orbit for a planet at the point of zero gravity, because the planet could be a binary planet as above. I guess there would be no orbit, but maybe a meta-stable point, where the planet ( or system) would hover. Meta-stable because it would be easy to be attracted to one or the other star from a small perturbation due to their extended masses.
