iCloud doesn't allow me to backup
In order to fit within the 5GB limit for the iCloud backup, you need to have the used space on your device be less than 5GB. Technically, Apps don't count against this, although their documents/data does. There are other items that don't count, as well, you can read more about iCloud backups here.

You stated that your iPod had 5GB available, but the space used for the backup is the 5GB that you have in iCloud, it is not on your device. 

If you have more than 5GB used, as is likely given that it has told you there is not enough available storage, then you will need to either pay to add more iCloud storage, or just use iTunes for free. 

If it is your first time backing up to iTunes and you have lots of data, it will likely take quite a while, so you should be patient when letting it complete that first backup.