I have a few repository classes that are meant to talk to different kinds of data, deriving from an IRepository interface.
In implementations, the code talks to a data source, be this a directory of XML files or a database or even just a cache. Is it possible to reliably unit test any of these implementations? I don't see a mock implementation working, because then I'm only testing the mock code and not the actual code.
No, you'd use a mock when you were writing a class which uses an IRepository. For the implementations of IRepository, you'd need to test against the appropriate data source. For databases, that's a bit of a pain - for a file system, slightly less so.
Where possible, if you can express your implementation in terms of streams or readers, you will make your life easier: tests for those parts of the implementation can go against in-memory data sources, or streams from resources in the test assembly. Of course you'll probably need some tests which go to a real database or the file system, but hopefully fewer.
Whether you'd call such tests "unit" tests or not is a matter of how you define unit tests; personally I don't care too much about the names involved, but I do care about having tests. For databases in particular, these can be somewhat painful (especially if you want to be able to run tests in parallel) - but they can also be incredibly valuable, in my experience.