As each kid/family/situation is different, it's hard to do a fixed age.
We've done one night away (in the next town over, ~30-45 minutes away) when the kids were 14-16.
An extended stay, out of state/country, kids would need to:
A) Have transportation. In our area, that means they have their DL (at 16.5) and some more experience behind the wheel. But, we don't live in an area with public transportation.
B) Be responsible/trustworthy. Presumably parents have instilled that in them already and have already been in situations where they could showcase that.
C) Understand some basic home facts (where to turn off water/electric).
D) Have somewhere (friends/family) they can contact and/or go to if they do have issues (see A above).
We would also notify neighbors that we're out and to just "keep an eye" on the place... basically what parents (at least mine did) before wifi cameras.
When you put all of that together for us, it would come down to kids being 17. At that point, they're a year (or two) from potentially being out on their own (probably in college). And B & D is really not much different than being in a dorm.
What works for one family doesn't necessarily work for others.