In addition to connecting to iTunes (which is a great idea to back things up since when you run out of space, bad things can happen) you can log into iCloud.com and verify that your cloud storage and deleted items match what the device is telling you.
If there are items to come down from the cloud, you may have to clean it up there if the device has no room to move. It could be just wedged.
I would go to usage under the settings app and delete some third party apps that use a lot of storage but that you can recover (say games that have saved state elsewhere online or don't need saved state) and then power down the device.
Clean things up online using a computer and then plug into iTunes if you can borrow a PC or Mac from a friend (set up a new user account - don't connect to their iTunes and mess up syncing further).
Worst case, ask a follow on question detailing the usage stats and iOS version and we'll see what to do next.