Yes, it was / is a real car.

The DeLorean was completely built in Northern Ireland,  between 1981 and 1982, with 3 model years – 1981, 1982, and 1983. The company came and went before Back to the Future was released, and the movie production bought and used the cars without the knowledge of even John DeLorean himself.

The DeLorean was built for the North American market, and there was a dealership network with full Service Centers in all 50 states.