"A múzeum bejárata nagy és zárva van."
Translation:The museum's entrance is large and closed.
In English we'd either say "the museum's entrance" or "the entrance to the museum". "the entrance of the museum'" would certainly get the message across, but something would sound vaguely odd to the listener.
That "van" is a huge surprise for me – I understood that it should be omitted. Why is it present?
Because of zárva. The -va/-ve suffix transforms a verb into an "adverbial participle", and as an adverb it needs a verb to refer to.
It is the entrance of a certain museum. "The museum entrance" would be "a múzeumbejárat" or "a múzeumi bejárat".
We would only say "The entrance of the museum" if we wanted to stress "entrance" as opposed to, say "exit". "The museum entrance" implies a specific museum which has a single entrance. Unless, perhaps, we were talking generically about museum entrances.
I would say "the museum's entrance" to make it clear that I'm talking about a certain museum. That's also a possessive structure that matches the structure used in the Hungarian sentence. "The museum entrance" is more similar to a compound noun, like múzeumbejárat. But I just might be nitpicky. I do see your point, and in actual usage the difference wouldn't matter much.
I can't see much difference in this case, although in, for instance, prison guard the composite noun has a somewhat different meaning, being an occupation, and so "the prison's guard" and "the prison guard" are somewhat different.