At the foot of the Campidoglio, next to the Aracoeli staircase, you'll see the ruins of a 2nd-century apartment block (insula). The ground floor is thought to have served as a tavern, with four upper storeys given over to housing.
To visit you'll have to book a guided tour in advance (by phone), but you can get a pretty good idea of the cramped, squalid conditions many ancients lived in just by looking in from the street.