I'm going to assume you are in the United States.
If you have a garage and some spare monthly income for parts then the S14 is a great option. You should be able to pick up a decent one(they are all going to be beat to hell) for a few thousand dollars. Then you'll proceed to strip it down and rebuild it with fresh parts. The SR-20 motor swap, rear axle/LSD swap out of the J30, and the 300ZX TT brake swap are just a few of the things you'll want to do.
In the end the S14 is a cheap econobox with a ton of performance parts swapped in to it. It is going to creak. It is going to rattle. However, if treated properly it is going to be stupid fast. Depending where you live there might still be an active 240sx tuning community. It is definitely a dying platform though as the few remaining 240s are wrapped around poles and slammed in to walls.
The Mazda Miata is similar to the S14 with a huge community and a gigantic parts list available. Don't let the stigma of it being a "girly" car fool you. Mazda has put a ton of research in to making it one of the best handling cars on the planet. All it needs is 200-250whp and it's a go-kart.
I like the 240SX platform, but finding one that isn't riced up or beat to hell (like there's a difference lol) is getting hard. My buddy in Houston, TX has/had the fastest KA powered 240 for a while. Not sure if the record has been broken since then. Now he has a 2JZ in and is a lot faster.
@Patrick: There's nothing wrong with Honda. They have some really well-engineered engines that are often years ahead of anybody else. Ex. the B16A motor, which came out in the 80s, has port flow numbers that weren't matched by anything domestic until the LS1 (which coincidentally have almost exactly the same flow #s as a Mitsu 4G63 head, 1st gen). Honda K-series have made close to 700whp without so much as removing the valve cover. They feature fully bed-plated bottom ends, and a head that flows nearly 18% better than even a fully race-ported B16A head. And Honda F20/F22 (S2000) head flows even better than that.
Living in TX, there was a whole Honda group that was gunning for the big single Supras -- specifically SW. I don't think anybody could ever really touch him short of T1 Tony, and as a whole we weren't successful.. BUT that left a lot of really REALLY fast Hondas. But alas, there's only so much you can do with a FWD platform. Highway roll-racing was the soup de jour, and for that you need big HP numbers. It's easy enough to make 800whp+, but it's another thing to actually put it down to the ground, even with tires.
I just need Bitcoin to take off so I can get one of those Underground Racing Gallardos. Seriously, 1500whp+ in a platform that weighs little more than an Integra? SHUT UP AND TAKE MY MONEY!