The problem

Are there any theoretical limits on who can be appointed to the Supreme Court, or can one in principle appoint six babies to the Supreme Court?

That would be Article III. Don't see a minimum age. A great deal about the Supreme Court is based on precedent; notably, there's nothing in the Constitution that gives the Supreme Court any particular authority in deciding what is and isn't constitutional. As I recall the story (you've been warned), Chief Justice John Marshall was responsible for that; essentially, Congress filed suit against the President on grounds what he was doing was unconstitutional (might have been Jefferson applying military force without a declaration of war, but I'm just guessing), and if the Supreme Court had said the President wasn't allowed to do that, the President would have done it anyway and things would just have deteriorated into an even worse constitutional crisis; so instead, the Supreme Court said "we do have jurisdiction to decide what's constitutional, and our decision is that what the President is doing is allowed". The President wasn't going to object to the ruling since it was in his favor, and thus the President effectively supported the Supreme Court's claim to have jurisdiction to decide what's constitutional.