Well I had a pet jackdaw for a time, he came to me via our local castle where he was frightening the visitors by swooping down and trying to steal their ice-creams, and he was young when I first had him - however he resented living inside and greatly preferred to find a roost for himself in the garden and then come and knock on my window when he wanted breakfast.
There's no logical reason why you shouldn't have a pet crow that I know of, but how you get your hands on one is another matter. We have plenty stop off in the garden each day, but they nest in the woods at the back, and they nest very high. You certainly wouldn't be able to buy one in a pet shop.