The "Android Private Libraries" item is missing from my project (it doesn't appear in the Project Explorer in Eclipse)... It appears as though it isn't getting built into the project either, as I'm getting various errors regarding jar files in my "libs" folder...
It does appear in Java Build Path (yet it appears to be empty - no jar files appear under it in the Libraries tab), though, and I did make sure it is checked there.
If it is of any relevance, the project at hand has a lot of Android library project dependencies as well, and in those library projects the jar files under libs/ do appear under Android Private Libraries.
I found the problem - I just recently added a new library project as a dependency, and it had an earlier version of android-support-v4.jar in its Android Private Libraries than the one I was using, so that ended up causing the entire Android Private Libraries folder not to be generated in the end project.
Found this out by looking at the Android Console after trying to build.