Say, if I want to store PDFs or ePub files using MongoDB's GridFS, is it possible to perform full-text searching on the data files?
You can't currently do real full text search within mongo: http://www.mongodb.org/display/DOCS/Full+Text+Search+in+Mongo
Feel free to vote for it here: https://jira.mongodb.org/browse/SERVER-380
Mongo is more of a general purpose scalable data store, and as of yet it doesn't have any full text search support. Depending on your use case, you could use the standard b-tree indexes with an array of all of the words in the text, but it won't do stemming or fuzzy matches, etc.
However, I would recommend combining mongodb with a lucene-based application (elastic search is popular). You can store all of your data in mongodb (binary data, metadata, etc.), and then index the plain text of your documents in lucene. Or, if your use case is pure full text search, you might consider just using elastic search instead of mongodb.
Update (April 2013): MongoDB 2.4 now supports a basic full-text index! Some useful resources below.
http://docs.mongodb.org/manual/applications/text-search/
http://docs.mongodb.org/manual/reference/command/text/#dbcmd.text
http://blog.mongohq.com/blog/2013/01/22/first-week-with-mongodb-2-dot-4-development-release/