I mean so good that one person could truly impersonate another, even to those who know well the other person.
Maybe if the idea is downgraded to just trying to get past a security camera or guard station some such masks would be feasible.
I know MI and MI2 were movies, and I understand “suspension of belief” and all that. I just wonder how far along current technology could go along these lines.
Sure, but it would start melting after a few hours. 
Seriously though, you could probably make a mask that looks a lot like a person. My problem with it is that I don’t think it would fit right on someone else’s face. If the person I want to copy has a small nose, I wouldn’t be able to wear their mask because my nose is frickin’ huge. It would be quite obvious that I was wearing a mask.
Just opinion, I don’t know anything about what kind of technology is neede.
If you are asking if it is possible to make your face look 100% like another person’s face then I think the answer would probably be no. As mentioned by c_goat there may be too much difference in bone structure, nose and the like to allow a convincing facsimile to be made. Of course, if you look reasonably similar to another person already you probably could pull it off (maybe a sibling who would likely have similar features to yours).
If, however, you are asking if a convincing mask could be made to just make you look like someone different (but not necessarily a particular person) then I’d almost definitely yes. I’ve seen a bunch of shows on Hollywood makeup artists who turn actors into different people. They are quite adept and could almost certainly make you look completely different and no one would be the wiser except perhaps upon very close inspection.
There was a fascinating series of programmes on BBC a while back where they made up people to change their racial appearance; for example, a black guy was made to look white, then they sent him into places where white men don’t go (and places where black men aren’t welcome).
It wasn’t entirely convincing (although that’s easy to say, because the viewer watching the programme I already knew the score), people were taken in, but something was just not quite right about the face; I think it was the restricted range of expressions.
Not only would your face have to be the same shape to pull it off, you will likely have to be about the same size (height, weight) to pull it off also.
I’m 6’5" and if I were expected to be someone who is 5’8" I think it would be hard to pull off.
It might be easier to make a short person taller rather than the opposite.
There was a fascinating “Saturday Night Live” sketch with Eddie Murphy that went along similar lines.
Come to think of it, Lenny Henry did a show where he was made up to look like Eddie Murphy, Richard Pryor, Steve Martin and a whole bunch of other people; I seem to recall that it was quite convincing, although maybe my memory is just being kind.