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It will not affect system stability, but... Surge protectors do not work at all without a ground wire to drop excess voltage to. Any kind of line voltage disturbance could kill every device.

Additionally, without any ground wire to pull the housings of devices to ground, the potential for a short to energize the case and then electrocute you is also high.

additionally additionally, if you have grounded outlets that don't actually have a ground connection running to them, that means either the wiring system is broken or it was "updated" by an unlicensed hack job who has undoubtedly made numerous more dangerous decisions elsewhere in the circuit.

If your house is entirely ungrounded you really should have an electrician come update it ASAP. Outlet grounds have been mandatory since 1971. The chances are high that wiring predating that code is still using old cloth-wrapped wire insulation or even knob&tube, both of which are huge fire risks as the insulation is decayed badly by now. It's expensive to have all new wire pulled but it is necessary.

Permanent damage to PSU? Permanent damage to other components?

Or worst: permanent damage to yourself.

Without a ground there is nowhere for a surge to go, permanent damage is much more likely. Surge protectors or a UPS will not protect against surges at all without a ground.

There's also no ground so the chassis may have enough voltage on it to cause a shock if you touch it. This could also damage components as they are not grounded and touching things can introduce high voltage from static electricity which will have nowhere to go.

Additionally if you have ethernet connected to it the system may end up grounding itself through the ethernet cable, if the device at the other side does have a ground, which could cause issues.

So it basically just means you have a much higher chance of damaging the parts, or injuring someone touching things.

I know that the answer is yes

I mean, there you go, and all of the above. I'd add in a pretty large fire risk too. I hear my battery backups kick in regularly, and we're talking about enough power to equal a large appliance (at least in my case). It's 100% worth it to move them to a grounded outlet.

Electric shock to you? Get an electrician to check it and sign it off?

Other question for anyone listening here: How can we check if an outlet is correctly grounded ? I live in a faily new appartment with good outlets, but maybe I should make sure they are OK ?

I mean it depends on the intensity of the surge, but basically you'd be making it so your PSU is unable to protect the devices from surges. The more sensitive the electronics, the more critical the ground is and CPUs are pretty darned sensitive among other things. And depending on the type of components in the PSU, "surges" also include things like inrush current. Basically, when you turn on a transformer or certain other devices, there is a surge of sometimes as much as 10 times the rated current to create the initial magnetic flux. Depending on the components, this excess energy may end up getting shunted to the ground to avoid pushing it through your electronics. So if it can't do that, you likely will blow fuses a lot when switching the power on (hopefully there are fuses), or if you're touching the case which is supposed to be grounded, you may end up getting that jolt.

Anyway, without grounded outlets, and especially if your electronics are cheaply made because many expect there to be grounding and don't build in extra components to deal with not having a ground, you are likely to significantly reduce the life of your electronics, your life, or start a fire without even considering major surges. If you have a high-end PSU, you may never have a problem until that surge happens. How stable is your power? Because even a normally small surge combined with a cheap PSU, and no ground, is pretty likely to end up in damage to electronics at the best case.

Do you have money to replace everything plugged into those outlets, and sufficient home insurance that also ignores such things? Then, no, I guess.

Just take an hour and make a ground yourself. It doesn't take a lot of specialized knowledge to do so.

Edit to say, I'm pretty sure any surge protector worth itself has a ground output on it already. Just run a wire from it into the literal ground if possible, or over to a place in your home that is properly grounded. You're just trying to give something like a lightning strike a path of least resistance to discharge into. Any metal conduit in your home SHOULD be grounded, so that's an easy option.

1. Install GFCI plugs. [^1]

2. Buy a small, appropriately sized UPS.

3. Profit?

[^1]: 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

Disclaimer: Am not actually electrician. Just know this about GFCI from electrician friends and pointing to a decent explainer.

You don't need to have wired across the room. You can put them through the wall like every other cable. If the wire tubes are not full, it isn't very complicated. I put my Ethernet wires in the wall.

Run the ground into the ground. You can get metal wire that would work nicely.