A German preposition, which we've borrowed in a few set phrasal verbs of that origin.
Whenever you come across it it's best to treat it as a part of the phrasal verb, rather than trying to translate it on its own. It seldom carries any discernible meaning of its own, but rather modifies the meaning of the "base" verb. It's also used as a verb prefix.
å gå an = to be possible
You can find more examples and information in this thread.
They're interchangeable when something really is impossible to do, but "det går ikke an" can also be used when something is possible, but socially unacceptable, for instance.
Without the negation, "det går an" means that it's possible to do something, while "det er mulig" can be used about possibility in a more general sense, and thus translate to both "it's possible" and "possibly", depending on the context.