If I understand your issue correctly (and others can correct me if I'm wrong), a backup will not restore your iPhone.
You can connect your iPhone to a PC/Laptop/MAC and iTunes will make an automatic backup. You can also manualy save your files, to make more space on your iPhone.
Once you've saved your stuff, you can delete them from your iPhone if you want, to make more space.
You're running out of space. It's not a big fault or anything. Don't worry. You can make space.
The message regarding storage full is coming from iCloud as you are backing the device up to iCloud.
Connect it to a computer and backup via iTunes... no need to purchase additional storage.
A backup will not restore the device but can be used at a later time to restore the device.
Space available on the device is irrelevant to the backup process.