If a container like a balloon but with constant volume is filled, is it possible to pack air molecules so closely together that they don't have any empty space between them? If so, what would this material look like?
Short answer: No.
As you pump more and more air into the container, the pressure rises and rises. At some point, the molecules are so close to each other that instead of a gas, you get a liquid. If you continue even more, eventually you'll get a solid.
In this solid, atoms/molecules are arranged in a regular pattern with well defined distances between them (which depend on the pressure), but those distances are not zero so technically there's still "empty" space between them.