In the last footnote of the story, the narrator presents us with an idea that totally inverts our understanding of the Library. (For more on this, see our discussion of "What's Up With the Ending?") Another librarian has pointed out that the entire idea of the Library is pointless. Why not just have all that information contained within one book? Of course, that book would have to have an infinite number of infinitely thin pages, but that's theoretically possible, right? And then space wouldn't be such an issue – you wouldn't have to find room to keep all those volumes.
Is it just us, or does this idea remind you of the Internet? In the digital age, we are now capable of storing near-infinite amounts of information in a negligible amount of space. So the "infinite book" is a symbol of the Library (which is itself a symbol), but it's also one of those neat ways in which Borges imagined a model of digital culture. (Intrigued? You might want to take a look at this New York Times article about Borges and digital culture.)