Should I even "consider" training for a medical career since I am an ex-felon? Would there even be any jobs available to me?

I don't think that anyone should ever give up on their dreams of a career. However, I am currently in a medical assisting program and just to get into the program, they do a background check. I am not sure what they specifically look for but I would really try anyway. I also work for a hospital and they Don't hire people with a record.
The healthcare industry is broad and has many options. I would look into it further. Even though you might not be able to work for hospital or doctors office, there might be nursing homes,home health, clinics,etc.

Don't GIVE UP!