That is, say you were wandering Rome at 2am for some reason. Could you visit the baths, or any of the various temples? Could you just walk in to the Colosseum? Or did things like that have "operating hours," the way a park today might be closed dusk-to-dawn?
Normally Romans went out to dinner at someone else's house and then went home with the sun and went to bed. As a side note if you were staying in Rome you were not likely to leave your house really after the sun went down. Rome was a rabbit warren with no street lights and it was easy to get lost/mugged/killed. Further, after an edict promulgated by Julius Caesar all deliveries of grain and stuff had to be made during the night to cut down on congestion during the day. This meant that the narrow streets were filled with jostling carts trying to deliver stuff. So the answer is no, places did not have specific opening and closing hours (aside from baths, which were most active in the mid to late afternoon, if you were not wealthy enough to have one in your home) but it would be highly unusual for a person to just be wandering about. That said I am not an expert, if anyone has more knowledge please correct me.
Sources: Daily Life in Ancient Rome, Florence Dupont.
The baths could be used until sunset and some had separated hours for men and women (Under Hadrian, mixed bathing was forbidden because of some scandalous behaviours). Women got worse hours for higher prices as well. But under the time of Alexander Severus, it appeared the baths were kept open all night.
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