Moodles have never been install it once and never tweak. Usages necessitates tweaks. So the 'service unavailable' screen might be apache or back end DB or both - which last only a few seconds ... users already logged on not apparently affected unless they just happened to click something at that time.
Turning on debugging may not show anything as the error you are describing indicates user isn't even getting to the site/moodle.
So, check error logs ... apache error logs as well as DB logs.
Consider installing MySQLTuner and see if DB could stand some tweaking.
What's 'top' look like? (top command from ssh session as root user)
'SoS', Ken