How to get rid of other storage space on iPhone 4s?

This is so frustrating. I've looked up how to do it, and it says to restore your phone to its original settings. Unfortunately, in order to restore your iPhone, you need to back it up first (so you don't lose all your contacts and stuff). Well, I can't back mine up because my "storage is too full", so I'm stuck in a never ending circle of things I can't do. I literally have 3 photos, I just deleted all my calls/voicemails, I have maybe 3 extra apps that don't come with the phone already… I've basically gotten all my storage down to as low as possible. The problem is the "other" category. HOW do I get rid of it? It's taking up like 2/3 of my storage. And the other problem is, when I try to take it to the apple store, they tell me I need to back it up before they mess with it so I don't lose anything. Well, I'm basically stuck.

Can't get rid of it entirely as it contains files essential to iOS. Could try a cache cleaner like Phone Clean, though that is a desktop app.

That "other" folder happens to be your OS. Removing that will render your handset as a brick.

"Other" holds things like system software, contacts, Messages data, settings and things that are not listed elsewhere in the capacities list. If you want to know how to delete the "other" files, this step by step guide will help you a lot.
http://www.mac-dvd.com/how-to-delete-other-capacity-in-itunes.html