I'm thinking of purchasing a NAS that can do RAID1, to have a local backup. Can such a device be left always on ? What happens in case of a power outage ?
To elaborate on @ta.speot.is answer - Yes, a NAS (or computer) can be left on 24/7 - there are arguments to be made that this is actually good for devices as they are not subject to the stresses caused by changes in temperature or the sudden rush of current on startup.
In the case of a power outage, you should ideally have a UPS which can be used to do a controlled shutdown - failing which, of-course the NAS will just turn off. (There might be a battery backed write cache to hold your data for a few hours/days, but I wouldn't count on it).
I suspect the real answer to the second question would be that a NAS will generally use a journalled filesystem (most filesystems are nowdays I think), so if there is a sudden power outage it can roll back a short while to the last "known good" write and you don't corrupt the filesystem.
In short, yes. I have an old Netgear ReadyNAS which is reasonably power efficient (versus a home-built desktop-as-NAS setup). If the power dies it will turn off. Some NAS devices can be set up to be always-on, and will come back online once power has been restored.
Be aware that RAID is not backup. Furthermore, electrical storms are a danger to the NAS and the drives it contains.