Periodic boundary conditions are an approximation that greatly simplify the analysis. The size of a crystal is enormous compared to the size of an atom, so on a small, local scale the system appears to be periodic. In practice this is a very good approximation. It does leave out the physics that occurs in the vicinity of the surface. That region is handled separately, and is the large field of surface physics.
As I understand it (not too well, I'm not a theoretician) there are technical problem with periodic boundary conditions related to the fact that the position operator is unbounded, and strictly speaking, inadmissible. Years ago I had a question similar to yours, and looked into it a little. I have a memory, perhaps unreliable, of stumbling across a paper where a quantum mechanical treatment of a finite crystal was presented. The author was Klaus Fuchs.