A bibliotheque in French is a library - a place where books and other materials are kept for people to read, study, and borrow. Among the options, une salle de lecture (a reading room) is the closest description of what a bibliotheque is.
This question tests an important false-friend distinction between French and English:
bibliotheque = library (a place for reading and borrowing books)
librairie = bookshop/bookstore (a place for buying books)
Many students confuse librairie with the English word "library" because they look similar, but they have different meanings. A librairie is a commercial shop where books are sold, not a public reading space.
The other options are clearly unrelated: un hopital is a hospital, and une ecole is a school. Neither describes what a bibliotheque is.
