Can you hook up extra storage box that's not microsoft to xbox 360?
Instead of buying the extra microsoft x-box storage, can you hook up any external hard drive to the xbox to increase the storage beyond 4 gb?
You can put up to two 16gb flash drives (as long as they are compatible, a quick internet search will give you a list of compatible drives) to any xbox for additional storage. These do not need to be xbox specific, but you do need to make sure they have good enough performance for the xbox to use them.
As far as hard drive, you can't just hook up an external hard drive to the xbox. Since you mention 4gb, I assume you have the new version of the xbox. The only Hard Drive that works with this xbox is the official MS accessory. The housing and connector shape are proprietary, so you need to get the official one.