I'm in the same boat, with the same budget.
Have gotten the advice to buy a class-qualified car, not just a track rat. Even if you never intend to run wheel to wheel, this is supposed to help resale down the road, and to give some confidence that the car was put together correctly.
So far, I've figured out that some cars qualified to run in Spec Miata, Spec E30 and Camaro-Mustang Challenge fit the budget. You might also be able to find a Spec 944. Seems that some E30/E36 cars prepped for various other classes could be had at or slightly above that budget as well. Each of these classes has a website with a classifieds section.
Different racing classes will be more or less popular depending where you are.
It's pretty unlikely that any of these will be street legal, so if you really can't spring for a towing setup, you may ignore the above. I'm fortunate that I already have a prime tow vehicle (diesel Excursion).