To clarify a bit more, were there any places where ordinary citizens (or nobles, etc.) could look at or read old documents or statues, pottery, paintings, etc.?
Please note that I don't mean museums in the sense of private collections that the owners only had access to, but rather that the public could see.
Thank you for your time.
Churches actually filled a lot of these archival/museum type roles for many communities. The public might not have had complete access, but the church provided a lot of protection for religious and some secular artifacts, manuscripts, and artwork.
Ennigaldi Nanno, a neo-Babylonian Princess who lived in the 6th c. BC is claimed to have created and curated the first known museum in this rather sparse wiki page it states that she labelled the exhibits and excavated some herself.
It doesn't make clear if this was a private or public collection, I would think more likely private, but I think it's worth mentioning as the whole digging up old stuff and sticking a label on it is very redolent of a modern museum.
If anyone has better sources for her and her Museum I'd be very interested to see them.
Some specific examples of medieval churches as museums: a meteor was kept at a chapel in Alsace-Lorraine, a giant’s rib (actually a wooly mammoth’s) in Wales, a stuffed crocodile at St. Bertrand-de-Comminges in south-western France. See J. G. Davies, The Secular Use of Church Buildings (SCM Press: London, 1968), 76.
One famous art gallery that comes to mind is the Pinacotheca in a wing of the Propylaea on the Acropolis in ancient Athens which contained paintings of famous battles (or so says Pausanias).
You might also consider [cabinets of curiosities] (http://en.wikipedia.org/wiki/Cabinet_of_curiosities) or [Wunderkammern] (http://www.moma.org/visit/calendar/exhibitions/272) beginning in the 16th century, which were private museums of a sort.