Your comment got me wondering about how widely Hellenistic cosmological models of the spherical earth circulated. I, like you, assumed it was only known to a small group of scholars like Eratosthenes. But digging around a bit, it seems like any literate person in the Mediterranean basin and Middle East during the time of the early Roman Empire could've been exposed to the idea of a spherical earth, via Aristotle or the Pythagoreans. Or at least so says Wikipedia's (surprisingly good) page for the history of the spherical earth model: "After the 5th century BC, no Greek writer of repute thought the world was anything but round." [1]
The page documents how this then filtered into Roman and early Muslim astronomical traditions. Granted, since only a small percentage of people in these societies were literate, it was probably true that most people in, say, the Roman or Sasanian empires thought the earth was flat. But given what we know about the ways that premodern books were read (often in public, sometimes to a large groups of people) it seems to me like an open question.
There's also this very interesting aside about the possibility that the Phoenicians also knew the earth was round, perhaps even predating the Greeks: "In The Histories, written 431–425 BC, Herodotus cast doubt on a report of the Sun observed shining from the north. He stated that the phenomenon was observed during a circumnavigation of Africa undertaken by Phoenician explorers employed by Egyptian pharaoh Necho II c. 610–595 BC (The Histories, 4.42) who claimed to have had the Sun on their right when circumnavigating in a clockwise direction. To modern historians, these details confirm the truth of the Phoenicians' report and even open the possibility that the Phoenicians knew about the spherical model. However, nothing certain about their knowledge of geography and navigation has survived."
[1] https://en.wikipedia.org/wiki/Spherical_Earth#History