Greek Helmet/ armour database

[Paullus Scipio]

AFAIK you don't, because these items belong to a museum, the museum owns the copyright.

.....I'm afraid this is incorrect, the museum may own the items, but it cannot own copyright in any photos taken by a person. That person is the originator/author of those photos/images, and hence owns the copyright in them. Ruben is quite right in this instance.

In the example you linked to, Royal Armouries has a regulation ( the 5th bullet point down) that says that they allow photography provided that it is not used commercially, or uploaded to the internet. This is a mere "wish" on their part, and has no real legal standing - you may notice that they request that visitors comply.
Another example of copyright from your example is that they don't allow photography in the cinema. That is because they are showing someone else's images ( and they are the copyright holder), and a person photographing those images ( that is 'copying' someone else's images, could potentially lead to a breach of the image originator's copyright, and Royal Armouries could find itself party to any legal consequences)
I would have no hesitation in providing any personal photos taken there to the Database, and indeed as the Database builds, inviting them and other relevant institutions to contribute relevant photos themselves, which I'm reasonably sure they would do, suitably accredited.

Incidently, I should add I am a qualified solicitor in England and Wales as well as here, with some 30 years experience here in Oz and in the U.K.