iPhone 5 not starting
Seems something has corrupted on the iOS and that is why it cannot boot properly. 

I recently had an individual I know face the same issue with an iPhone 7. They were able to take the phone to an Apple store and was able to fix the issue without wiping the phone, saving all data on it. Apple then performed a diagnostic to ensure the hardware was working correctly because it was still under warranty. 

If you cannot go to an Apple store, an iTunes recovery with your computer is necessary. Good thing you said you can put it in recovery mode because that's what you'll need to do, then use a computer to recover it. 

This is the only option you have without going to apple support. If Apple cannot manage to fix the issue with the method they used for the individual I know, they will end up doing a wipe and reinstall first to ensure this isn't hardware at fault, then proceed with a hardware diagnostic to confirm that. 

Keeping a devices memory so full that it cannot update is risky. Try not to do that if you can. Use iCloud storage or other services such as Dropbox, Google Drive, One drive, or MEGA to hold/backup your files. 

I highly also recommend always keeping your phone/computer updated at all costs. Security issues are happening all the time now.