Kit cars are still a thing: https://en.wikipedia.org/wiki/Kit_car . That said, they are almost all replicas of old cars, and in some countries getting them licensed for use on the road is hard.
From the Wikipedia article: "A kit car should not be confused with a 'hand built' car or 'special' car, which is typically built from scratch by an individual."
The parent, a few up, was talking about getting every part: to me that implies you're using pre-made body-panels, etc., rather than making them from scratch.
It wouldn't be cheap. The safest, cheapest and most efficient "packing crate" for the parts for a single car is the assembled vehicle. (It's somewhat different when you can stack 50 left front fenders.) A kit car (most use fibreglass bodies) would be a much better option. Expect to spend about 1000 hours (much of that building jigs). The second one you build goes much smoother and faster than the first, even if it's a different kit based on a completely different pair of vehicles (the "source", from which most of the drive train parts will be drawn, and "target").