You mentioned Git. That is exactly what you are looking for. Git and SVN are designed for source code for programmers, but there is nothing that restricts it to that. At my work our SVN server holds documentation, programs, pictures, etc.
At it's core, version control just syncs folders and provides a history of who did what. It really has nothing to do with code.
These articles show you how you can install SVN on your NAS:
http://www.visualsvn.com/server/ http://www.visualsvn.com/support/topic/00022/
Then you can install this program on all the computers that need to use it: