For $5000, you could buy an old school MG Midget. Super light, pretty original (cuz everybody drives hondas and toyotas, etc). My dad bought my MG for $500 at an auction. You can probably find a pretty cheap MG (around $1000 more or less depending where you find it), but i've heard that they're kind of hard to maintain. They have 1.5L engine or less, depending on the year. If you wanted to do an engine swap, you have a lot of room to work with. It's a really small car though, only 2 seats and maybe someone small can fit in the back. Go look up some pictures of it.
If you only spend about $1000 on buying the car, you can spend your other money tricking it out. That's what I would do. (engine swap!)
oh, also it's pretty good on gas for an old school car.