the only actual requirement I know of is that you should be born in America. It's notionally possible that the next election could be between Hillary Clinton and Condoleezza Rice; but I'm betting neither of them will be selected by their parties. If only one of them is (also unlikely), she will be beaten by a male.
you don't have to be stupid, though you'll probably need a thick skin. The current Bush seems a bit of a dimbo, as did Reagan; but Eisenhower was a successful general; Clinton was a Rhodes scholar; and even Nixon, though paranoid and petty, was a bright guy.