Существуют ли серьезные проблемы для домашнего сервера NAS, который слегка дрожит во время небольшого землетрясения?

In general that shouldn't really be an issue, hard drives are pretty vibration resistant. If you're worried though, you might considering doing what they did, on a smaller scale.

If its a small earthquake, you might be able to simply isolate your home server by setting it on top of something that would dampen or resist vibrations, and of course secure your drives properly. Laptop drives and SSDs are probably better at this than desktops. If you consider that a hard drive turns are several thousand RPM, the 'vibrations' from a earthquake are minimal.

Vibration isn't really the thing that would worry you - earthquakes would be relatively low frequency - this suggests 0.1 to 70 hz is what engineers test against- its G-Forces. I'd need to find a proper citation for this, but 350G would void the warranty on seagate drives. I'll assume that that's a good average to work off of.

Now, its not entirely accurate to refer to acceleration due to the earth moving as "G" forces - the correct terminology is PGA, or peak ground acceleration. Dropping your laptop should cause 1G while dropping, or 1 ms^2, and stop based on whatever speed its dropping at right as it stops, up to terminal velocity. I'll leave it as an exercise to a VERY brave reader to work out what the terminal velocity of a given laptop is.

Lets then assume a worst case scenario. Lets assume an earthquake that will level your building, but somehow due to the magic of sheer dumb luck, nothing fell on your server, and its sitting there, untouched in the middle of all that carnage, like a cartoon character. Unfortunately there's nothing on the wikipedia page for what total damage would be like, but lets assume X+ on the mercali scale. Bad enough. Acceleration would be equivalent to 1.24g, which is significantly less than what drives are rated for! In short, unless a building falls on it, a earthquake that would cause a HDD to fail would be quite literally a end of world quake.

If this still worries you, there's a few options that might help.

Isolation mount the nas - maybe on a thick sheet of rubber. Isolation mount the drives - there's products meant to prevent vibration to quieten drives that may work. Practically though, it should be fine. I had trouble of finding reports of mass drive deaths due to earthquake related vibrations.

We are not talking here about major end-of-the-world’s earthquake, but rather small ones that can provoke the server to shake a little bit during 15-20 seconds while turned on.

I wouldn’t sweat a setup like this. A shake every now and then should not hurt the NAS in any way. Just one piece of very obvious advice: Wherever you physically place the NAS make sure it’s not in some position to—let’s say—fall off of a desk or drop at any significant distance. The NAS being constantly shaken to the point it would slowly move towards the edge of desk and then physically crash on the floor would be my biggest concern.

And past any of that advice? If you are building a NAS be sure to use a RAID setup of RAID 5 or 6 if possible. And if that is not possible, then maybe just RAID 1 for basic mirroring. That way if a small quake happens and somehow does choke one drive, you at least know your data is safe. Be sure to keep at least one spare hard drive around so you can swap out the bad drive with a working drive quite quickly.