Honestly, I feel like the "
Little Free Library" project has an interesting concept as is, I would probably just branch off of their pre-existing idea.
A traditional library that's similar to or is managed by the local government requires a lot of resources. So you need a way to keep inventory, sorting the books, keeping the books in decent condition, figuring out how to get books returned to the library if they're taken out (like penalties), etc.
The concept of the Little Free Library project kind of solves most of these issues by having it be a small and manageable, easy to re-establish if it gets knocked down for some reason, the cost is pretty low if not nonexistent, etc., but the only real "downside" is that there's no real ownership of the books. You take one and leave one, and whatever that person does with the book they took is up to them now.
Otherwise, if you mean something more personal, I imagine you would just put up bookshelves (or just shelves in general) and have only trusted people borrow the media or manage a small community where maybe the media has to stay within the area.