Early Viper : 30K$ and no issues what so ever. Nice project car that you can keep on modifying.
Early Diablos are also nice.

 

Dreamtripper, I have a 1987 911 targa and absolutely love it. I took it to Farnbacher Loles to have some modifications done, and it's a lot more peppy than stock. The 87-89.5 911's have the G50 transmission, which is a vast improvement over earlier trannies. The car has a real visceral feel. I considered other "older" sports cars, but reliability or performance issues steered me towards the 911 Carrera.

If you go with an older 911 that is normally aspirated, install turbo tie rods and check to make sure the cam tensioners have been upgraded. The 87 Targa and 01 Turbo are very different Pcars. Both are a hell of a lot of fun to drive, and both can be exhilirating. You can pick one up fairly cheaply. Get a PPI if you can. It's also a great project car, and there are books you can purchase so that you can do most jobs yourself. Take the plunge and enjoy.