In regards to the city of Rome itself, at least in the Republic they didn't have museums per se, but they did have many displays of captured artifacts throughout the city in the form of spoils for the general purpose of demonstrating Rome's power. Perhaps the most prominent example that comes to mind is that of the rostra, the platform from which orators spoke in the Roman form, which was decorated with the beaks of ships captured in various naval battles, beginning with Antium in 338 BC (Welch 500). The word rostra is just the plural of rostrum, the ram of a ship, which were the parts of ships on display. So whenever you watched somebody speak in the forum, you saw several beaks of various ships from various enemies the Romans had defeated decorating the platform.

Private individuals had a lot to do with display of material culture and artifacts. Private residences probably had the greatest amount of artifacts and spoils on display, with the former perhaps being the closest thing the Romans had to museums. Republican citizens who had killed an enemy and stripped them of their armour often displayed them hanging outside their houses (Lendon 177). Private victory monuments were also common in the Republic (500), though far more famous examples obviously popped up in the Late Republic and Empire (tributes to Agrippa, Augustus, the victory columns of Trajan, Marcus Aurelius et al., Hadrian's arch, etc). Indeed by the mid-Republic, the Forum and Capitol had become places where powerful men created monuments to their military success.

Large amounts of spoils and looted artifacts tended to end up in public display, though a lot still ended up in private residences, prominently beginning with the sack of Syracuse in 211 BC (517). While the trend was well-established by this point, Mummius' sack of Corinth in 146 BC saw a huge amount of looted Greek art and objects end up in Rome, most of which ended up in public display in temples and porticoes, and some also ended up in personal house atrium's (517). Statues were particularly sought after (517-9), but other items, for instance spolia opimia, tended to be dedicated to and left in temples.

So basically, (in Rome, under the Republic) no they did not have museums, in that they did not have public buildings purely for the purposes of displaying artifacts and objects. However, the Romans had public displays of artifacts and material culture from defeated enemies on show throughout the city. You couldn't miss the rostra in the forum, any decent private residence would have foreign art on display, there were victory monuments everywhere you looked, in the Early to Mid Republic citizens displayed the armour of defeated enemies on the front of their houses, and you could always go into a temple and have a look at all the arms, armour, standards and other items that had been dedicated there after past wars.

Sources:

Lendon, J.E. Soldiers and Ghosts: A History of Battle in Classical Antiquity. New Haven: Yale University Press, 2005.

Welch, K.E., "Art and Architecture in the Roman Republic", from Rosenstein, Nathan and Morstein-Marx, Robert (Eds.) A Companion to the Roman Republic. Oxford: Blackwell Publishing Ltd., 2010, 496-542.

One of the more celebrated social practices in ancient rome was that of the funding of public art by the aristocracy and government workers. it was a sort of "giving back" program/social convention to the commoners by the wealthy. these were usually copies of previous artworks or commissioned (from sculpture to mosaics), and would be displayed out in the open throughout the city. i believe they had vaguely designated spaces that art would go, but not large gatherings for the pure sake of their historical value.