Sure, with a performance penalty. Some GPUs do support double precision, though again at a performance penalty. You might run into issues with the fixed function parts of the GPU being single precision even on GPUs that support double precision types in shaders, depending on how you do things. Also, shading languages probably don't have nice libraries for double single precision so you'd have to roll your own, and probably without niceties like operator overloading.

It's not required for space rendering, as once everything is in camera coordinates you no longer have any visible precision issues (as long as you are using a float reverse Z buffer). You just have to be careful to make sure your world to camera transforms are done on the CPU, and you break large objects like planets into small LOD chunks with their own local coordinate systems, which you need to do anyway.