They're called "hot rods", and there's always been a small market for them. Depending on where you live the legal requirements to get one licensed vary from "easy" to "impossible", but it won't be cheap.
It is only expensive if you make it expensive, car mechanics are not that complicated. It is only expensive if you either buy expensive parts or have somebody else do it for you because it takes time and they are taking on the liability of any modifications they did along with licensing requirements to do such things commercially.
Home built vehicles can be registered just about anywhere as long as they have headlights, taillights, and a license plate light.