The British archaeologist Leonard Woolley actually found what is believed to be the oldest museum ever discovered while excavating the Mesopotamian city of Ur. It was created by Princess Ennigaldi, the daughter of the last emperor of the Neo-Babylonian Empire around 530 BCE.
The reason they were able to figure out that this was a museum was that they excavated a large variety of objects from different eras and areas in one place (much like one would find a modern museum).
[Long description of objects]... and we must needs conclude that the other objects found with it, whose presence all together in a Neo-Babylonian room it would be impossible to explain otherwise, are the remains of a museum of local antiquities."
Ur Expeditions Volume IX, Leonard Woolley
A great article from io9 quotes from Woolley's book Ur of the Chaldees describing the findings:
What were we to think? Here were half a dozen diverse objects found lying on an unbroken brick pavement of the sixth century BC, yet the newest of them was seven hundred years older than the pavement and the earliest perhaps sixteen hundred.
They even found what appears to be a museum label describing an object to be displayed (you can see it here at the top right of the page).
Ur of the Chaldees about the label:
"These," it said, "are copies from bricks found in the ruins of Ur, the work of Bur-Sin king of Ur, which while searching for the ground-plan [of the temple] the Governor of Ur found, and I saw and wrote out for the marvel of beholders."
The scribe, alas! was not so learned as he wished to appear, for his copies are so full of blunders as to be almost unintelligible, but he had doubtless done his best, and he certainly had given us the explanation we wanted. The room was a museum of local antiquities...and in the collection was this clay drum, the earliest museum label known, drawn up a hundred years before and kept, presumably together with the original bricks, as a record of the first scientific excavations at Ur.
EDIT: For those with JSTOR access, here is an article I came across from Fieldiana: Anthropology which briefly contextualizes Ennigaldi's museum in the history of the development of museums.
Egyptologist Claude Traunecker and the Strasbourg Uni team have been excavating Padiamenopet's tomb for years now. Theban tomb 33 had the reputation to be a dangerous labyrinth, but their findings tend to indicate it was conceived like a museum for future generations. In particular, there are a lot of really ancient texts that have been transcribed on the walls and there are addresses to the visitor encouraging him to study and copy them.
You can find more explanation and a small bibliography here on the dig's webpage. There also have been several documentaries on the subject.
Edit: sorry, I forgot to specify, it's mostly in french.
Roman temples in different times and places would often function as museums--Natural History museums in fact!
Adrienne Mayor in her fantastic work The First Fossil Hunters: Paleontology in Greek and Roman Times, describes how:
The bones of gigantic beings were treasured as relics of the mythic past and displayed as natural wonders in temples and other public places.(9)
and
No expense was spared to obtain up-to-the-minute natural marvels and old relics. Emperors and entrepreneurs began to plunder the collections that had accumulated over centuries in temples. Those sacred museum inventories included foreign armor and antique weapons, pickled mermen, celebrities' jewelry, the egg that Leda laid after her affair with Zeus in the form of a swan (probably an ostrich egg), great ivory tusks, the Golden Fleece, strands of Medusa's hair, gigantic snake skins, and myriad other miscellaneous mementos, including the bones and teeth of heroes, giants, and monsters. People brought curious objects to sanctuaries, not just as private offerings to gods, but in the spirit of creating communal museums where men, women, and children could contemplate natural wonders and try to puzzle out their meaning.(147)
The Greeks and Romans were very aware of fossils and understood them as physical remains of a mythical past--which she argues , is in part where the myths of giants, enormous heroes and fantastic beasts originated. These myths are in part attempts to contextualize the bones of mammoths, mastodons and other ice age mammals buried close to the surface throughout the Mediterranean basin. In that spirit, the peoples of the ancient world would use bones displayed in temples in much the same way that we use Natural History Museums today--to explain how the present world came to be through objects imbued with a sense of past antiquity.
Even better, Classical authors were entirely self-aware about this process(See, Mayer, The First Fossil Hunters, 260-282 for a full appendix of primary sources on the topic):
JOSEPHUS (b. A.D. 37)
In the area around Hebron (Israel), the early Israelites wiped out "a race of giants, who had bodies so large and countenances so entirely different from other men, that they were amazing to the sight and terrible to the hearing. The bones of these men are still shown to this very day, unlike any credible relations of other men.
Jewish Antiquities 5.2.3.
AUGUSTINE (A.D. 354-430)
Some people refuse to believe that [in previous ages] men's bodies were of much larger size than they are now. . . . In those days the earth used to produce larger bodies. . . . As for the size of the bodies, skeptics are generally persuaded by the evidence in graves un- covered by the ravages of time, the violence of streams or various other occurrences. For incredibly large bones of the dead have been found in them or dislodged from them. On the shore of Utica [Gulf of Tunis, Tunisia] I myself, not alone but with several others, saw a human molar so enormous that, if it were divided up into pieces the dimensions of our own teeth, it would, it seemed to us, have made a hundred of them. But that molar, I should suppose, belonged to some giant. For not only were bodies in general much larger than our own, but the giants towered far above the rest, even as in our own time, some few far surpass the size of others. Pliny the Elder, a man of great learning, declares that, as the world advances more and more in age, nature bears smaller and smaller bodies; and when he mentions that even Homer often regretted this, Pliny does not ridicule such statements as poetic fictions, but speaking as a recorder of the wonders of nature, assumes their historicity. But, as I have said, the size of ancient bodies is disclosed even to much later ages by the frequent discovery of bones, for bones are long-lasting.
City of God 15.9.
CICERO (b. 106 B.C.)
When King Masinissa [of Numidia] landed on the headland of Malta, his admiral stole the special tusks of astonishing size from the ancient Temple of Juno.
Against Verres 2.4.46.103.