I think the majority of astronauts, at least in the US, are either doctors, engineers, or pilots. As for aerospace medicine, there are only 2 civilian residencies, and only the UTMB one focuses on space, other in ohio. It's a 2 year preventative medicine residency w/ MPH or combined with IM for 4 years.
Planning on applying myself. We are about to hit space race 2.0 China and India are launching stuff to the moon, and multiple countries are drumming up to extend their presence in space. Public companies like SpaceX are starting up for the first time and contract out everything to governments. Quite exciting times.
BTW, the main book for the field is Fundamentals of Aerospace Medicine by DeHart, now Davis. A really fun read! Really cool if you like physiology -> altered gravity, pressures, gasses, etc. and stuff like radiation, life support systems, logistics. Oh yea, both the johnson and kennedy space centers offer SubI's!!!