Who can be buried in Westminster Abbey?

Nobody anymore. When I visited there some years ago, I was informed that a law passed in 1920 forbade any new graves in the Abbey (or in the entire London metropolitan area), which is why when one looks for, say, the court composer Ralph Vaughn Williams in the Abbey, one finds only his ashes (a few steps beyond Charles Darwin's grave, and a bit behind Sir Isaac Newton's). The last person to be burried in Westminster Abbey was the Unknown Soldier from WW1.

That said, the variety of people burried in the Abbey is a very wide one. The link below will give you a full comprehensive list of names of people whose final resting place is within its walls.

https://en.wikipedia.org/wiki/Category:Burials_at_Westminster_Abbey

The Dean of Westminster must give his permission for all burials and monuments in the Church. Ashes only are permitted. People who have served the Abbey in an official capacity, such as a Dean, a Canon, Organist or Surveyor of the Fabric may be buried here and eminent persons of British nationality from various fields may be considered. The last poet interred was John Masefield in 1967, and Laurence Olivier, actor, was buried there in 1991.

http://www.westminster-abbey.org/archive/faq/faq/what-are-the-main-dates-of-building4