"The Library of Babel" is a story about ideas – it's not really about the plot. In fact, there's not much plot to speak of. Using a Classic Plot Analysis to interpret it is kind of a stretch, but it's a fun exercise, so we'll give it our best shot. What do you think? Can you do better?
The Library is a series of hexagonal galleries, and we get the grand tour. Eventually we find out the most important fact about the Library – that it contains the total number of possible combinations of the 25 written symbols.
The meaning of life and the answer to life, the universe, and everything are out there in a book somewhere. Thousands of librarians rush out to seek their own fortunes.
And since nobody can just go on Google, it's really hard to find things. Chaos and violence ensue. Everybody gets depressed, and no one is any closer to discovering the meaning of life.
This is one of the narrator's big epiphanies (or realizations). See, EVERYTHING in the Library has meaning, even the seemingly random assortments of letters. For every word that looks like gibberish, there's a book in the Library that decodes that gibberish and gives the word meaning.
It really looks like the human race might die out any day now. The narrator keeps writing to distract himself from the sorry state of the mortal librarians, and ponders the immortal nature of the Library instead.
This is the narrator's second great epiphany. In trying to explain how the universe can be infinite, even though the number of books it contains is finite, the narrator comes up with the idea that the Library is "periodic" – in other words, that it repeats itself. He's like the Library's own Galileo, speculating that a traveler who walks far enough in one direction will eventually find himself back in the same spot.
The idea that the universe is periodic finally gives the narrator a sense of peace. Sure, he's alone and nearing death, but he consoles himself with the thought that there is some order – The Order – in the universe.