ReadyNAS NV+ v1 startet nicht/wird warm/schaltet ab

That it will only start when disconnected from the network smells (no pun intended) like a power supply issue.  The NIC and power-on circuit both are powered by a small section of the power supply that produces +5VSB (5 volt standby) power.  That power is always on if the NAS is plugged in, so you should probably be unplugging the unit as well as keeping it off.  Because it is always on, failure of the +5VSB is common in older units (and computers, too).

If you have access to a "spare" standard computer ATX supply, an adapter can be made to connect it in place of the internal NAS supply and see if that fixes it.  It's cheaper than buyng a supply and then finding out that wasn't the problem.  Then, you can either use that external supply to power the NAS while you move your data off to a new NAS (IMHO, your best option), or you can get a replacement supply (though not from Netgear).  Buying a replacement used NAS is another option, but you might just be buying somebody else's problem.