My iPhone 5 quit ringing. I checked settings and the side switch. What now?

When you go to Settings > Sounds > Ringtones and tap on a ringtone, does it emit a sound then? If so, Go to Settings > Do Not Disturb and make sure it is OFF.

Next steps would be to back up your iPhone, then go to Settings > General > Reset > Reset All Settings.

If the issue continues still, restore the iOS. Last step would be to take it in for service.

Locate a Service location (choose your country in the top right)