The Library of Babel
Occupation is clearly a major tool in characterizing the narrator of this story – in fact, we just go ahead and refer to him by his job title, "the librarian." He's really everything we think of when we think "librarian": he's bookish, scholarly, hardworking, and kind of fusty, but in a lovable sort of way. We picture him wearing a sweater vest and bifocals (his eyesight is failing, after all).
But keep in mind that everyone in the Library is a librarian. In fact, "librarian" in this case is just a substitute for the term "human being." So in that sense our narrator is really an "everyman" character – he's just like you.
Family Life/Sex and Love
Our poor librarian is a lonely, lonely guy. He has no friends, no ladylove (or gentleman caller, for that matter), and very few neighbors. In fact, sometimes he wanders the Library for days on end without coming across a single soul. The librarian's loneliness is one of the motivating forces in his life – he writes this story in order to distract himself from his solitude and the knowledge that the human race is on the verge of extinction.