There is no problem with gravity. We have general relativity to describe it in full, and it works as far as we can tell.
The "problem" in the movie, as I understand it, was that it was infeasible to evacuate everyone from Earth. It takes an enormous amount of energy to get even a small amount of mass out of Earth's gravity well. The movie called for some futuristic ability to manipulate gravity so as to make this evacuation possible. Note such an ability is pure science fiction with no basis in reality.
Now as soon as you mention "problem" and "gravity" in the same sentence, everyone will immediately start thinking about how general relativity and quantum mechanics don't play well together. This is a real "problem" that many physicists work on, but note that in the movie "solving gravity" could be done even without "solving gravity + quantum mechanics."