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?
Install GFCI plugs. [1]
Buy a small, appropriately sized UPS.
Profit?
Disclaimer: Am not actually electrician. Just know this about GFCI from electrician friends and pointing to a decent explainer.
GFCI plugs can be installed without a ground wire. https://diy.stackexchange.com/questions/225370/i-live-in-an-older-home-without-ground-wiring-is-it-safe-to-install-3-prong-gfc ↩︎
You are correct that this is technically in code and would protect against shock hazards in a neutral error situation but you also get the opportunity for the outlet to pop during the day when nobody is home and the battery to die.
We had a situation in our old house where someone who was technically correct but didn’t think it through had a gfci outlet upstream of the refrigerator outlet. Thankfully it popped while someone was home and we got everything corrected before we lost everything in the fridge.
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.