I know that the answer is yes, I should, but outlets near the setup are not grounded (even though they look like they are) and I don’t want to have wires running though my living room.
The real question is what are potential problems ? Occasional system reboots? Permanent damage to PSU? Permanent damage to other components?
For sure, a fridge is a really bad one to be using on an actual ungrounded GFCI, exactly for the reason of risk of expired food.
I like my homeserver but if something trips and its offline for a while it’s not gonna ruin my day.
The battery can be recharged eventually unless it’s already be discharged many times or it’s left alone and dead long enough to kill any ability to recharge it.