Were there places in ancient Rome (or Athens, Alexandria, or w/e) have places where they collected and displayed artifacts from the civilizations which came before them?
For example - It's obvious that the Romans fetishized Hellenistic Greek culture, which peaked centuries before the rise of Rome - Did they collect and show off authentic Greek artwork, or did they just appropriate the stylings and mythology for their own designs?
There may actually have been proper museums or antiquarian curio cabinets similar to how we understand them today, but the evidence is obscure and indirect.
Temples, sanctuaries, and public fora often acted as display places for Classical and Hellenistic Greek artwork brought back by victorious generals. Julius Caesar is said to have put a gem collection on public display in his forum, for example.
We know from Pausanias' Description of Greece that even in the 2nd c. A.D., Greek temples could be chock full of votive artefacts, artworks, and local curiosities, even to the point where it was difficult to see the cult statue or make one's way through the clutter.
Elite private households also had collections of antique statuary, vessels - essentially heirlooms, as well. We know from Vitruvius that there were libraries and private picture galleries, and it's conceivable that other display-worthy items (clocks, musical instruments, maps and armillary spheres, fossils, etc. (all things we know the Romans actually did collect, incidentally)) would have been kept there.
I'm not sure about collections of artefacts, but the Romans preserved and repaired the hut of Romulus for about a thousand years, presumably as a cult centre for the worship of Romulus.
Here is a great article about the likely excavation of a museum in UR. IO9 probably isn't the greatest source, but the author list his sources at the bottom. Really fascinating since the part they were excavating was from the 6th century BC. Crazy to think of people twenty five hundred years ago looking at artifacts from a thousand years before!