We started off using Dropbox. However, the limited space offered on the free accounts proved to be a limiting factor.
We switched to an open-source self hosted solution called
OwnCloud. We have it hosted on our own site at
http://cloud.team3309.org. It's not quite as polished as Dropbox but it provides the same functionality we need, with as much space as we can provide it.
The most important thing is to
establish good folder organization & naming conventions. We still have issues with students (and even sometimes mentors) who are in a rush and will create a ton of iterative CAD files without naming them properly and storing them in appropriate working folders, checkpoints, etc.
Also something to note: with Inventor, the lockfile stops the rest of the files from being synced if they are open in Inventor, so don't forget to close the program and allow the files to sync before you disconnect from the network.
Depending on your workflow, you could also consider using Github. It is pretty much the obvious tool at this point for software development, and while it isn't nearly as optimized for CAD and other non-source code files, the basic functionality is there.
Autodesk also makes the Autodesk Vault product, which we briefly considered using (they sent us a disc and license upon request) but it was too much of a strain on resources to set up a box and leave it running 24/7, find ways to access it from outside the school's firewall, etc.