I don't think "homeless" is actually a very useful description for the Roman world. The deal is that the vast majority of urban inhabitants were unskilled day-workers who lived a hand-to-mouth existence scrounging up whatever work they could. They survived by working on monumental building projects, at the Tiber dockyards, in the construction of roads and other manual labor, in minor unskilled suburban agricultural work, and in hawking. Their wages (provided they found work that day) were poor and paid daily. Harrison and others have pointed out that since rents were often paid daily and work was not always available in any given region of the city much of the city's population was probably essentially migratory, moving around in the city as working conditions changed. Inability to find work for a day might cause an individual or his family to be unable to find shelter for a night, and of course a large portion of the city's population was only seasonal, moving in and out along with the availability of work and the harvest. In a way, then, the urban population was to a large part semi-homeless.

Beggars, which we might consider the most abject of the poor, were habitually itinerant or homeless. Already in the Odyssey we see the ancient concept of the itinerant impoverished beggar, and already in Homer and Hesiod we see the issue with trying to identify something like homelessness from terms for beggars, in that the beggar is as much a literary and moral figure as he is a representative of a real group. Beginning with Hesiod and especially in Latin writers beggars are the result of laziness, the inability to maintain one's agricultural plot--interestingly, not long after Hesiod, Tyrtaeus was already referring to beggars as the result of wartime deprivation or other social crises, and lamented the beggar's loss of citizenship and social identity through no fault of his own. By the Roman Period beggars (either mendicus or pauper in Latin) had become firmly entrenched as a metonym for poverty in general, the same way that we find Cicero talking about omnis faex urbis when he means the entirety of the urban plebs. A few facts about Roman beggars stand out. That they were generally without purchased or rented shelter is usually assumed in Latin treatments--they often lived under bridges and arches, and Seneca specifically mentions a group of beggars habitually living under the wooden bridge of Sulpicius. Beggars tended to hang out along roads and crossroads (as the homeless often do in major cities to this day), and Juvenal mentions beggars pestering travelers near Aricia, whom he calls Aricinos as if his reader would recognize the reference. But there are lots of references to homeless beggars in antiquity, and the early Christian writers made the abjectly poor an object of particular concern--they're all over the Bible, after all. Several late antique emperors imposed legal prohibitions on beggary, making it an offense for men capable of working to resort to begging. But itinerant beggars were so common that indeed in the Greek world (and in the city itself) the Galli and Metragyrtae wandered from place to place begging alms in adulation of Cybele and Magna Mater