They buried the dead, and prayed.

Procopius' famous description of the plague gives a sense of its magnitude and horror:

"...at first the deaths [in Constantinople] were a little more than the normal, then the mortality rose still higher, and afterwards the tale of dead reached five thousand each day, and again it even came to ten thousand and still more than that."

In the face of such massive mortality, there was little that the imperial government could do. Justinian's only direct intervention was to ensure that the dead were buried. As Procopius says:

"[The emperor] therefore detailed soldiers from the palace and distributed money, commanding [his secretary] Theodorus to take charge of this work...[Theodorus proceeded by] giving out the emperor's money and by making further expenditures from his own purse, to bury the bodies which were not cared for."

Procopius goes on to describe the mass graves that the soldiers dug, and the desperate expedient of piling corpses in the towers of a fortress across the Golden Horn. John of Ephesus mentions that men were given substantial sums to haul away the dead, though it is unclear whether the imperial government was paying.

We know disappointingly little about the relief measures introduced after the plague. The Roman government frequently remitted (waived) taxes after natural disasters (earthquakes, etc.), and Marcus Aurelius is known to have excused outstanding debts to the government in the wake of the Antonine Plague. Justinian, however, seems to have insisted that taxes continue to be paid at the normal rates empire-wide, presumably because his finances (strained by his wars and building program) were in such dire straits.

Medical care, so far as it existed, was not an imperial responsibility. In the early imperial era, some Roman cities had salaried public doctors, whom any citizen could consult. Although a few Byzantine cities maintained these services, they tended to be discontinued as tax burdens mounted, and most had disappeared long before the plague. And of course, since the causes of the plague were not understood (Procopius blamed demons), there was little doctors could do for the sick. In terms of social security, the ball was in the church's court - and the church could only provide food and alms to those whose lost their families or livelihoods.

I talk in more detail about how the Greeks and Romans coped with earlier plagues in this video, which may be of interest.