Eratosthenes very famously estimated the Earth's circumference to a very accurately by measuring the difference in shadows in Alexandria and Syene, and this was around 240 BC. It's also well-known that mariners understood that the top of a ship's sails could be seen on the horizon before the ship itself could be seen, indicating the world is round.

Some very minor clarifications (but they're axes I like grinding) -- First: Eratosthenes used accurate measurements,* but there's no reason to suppose he visited Syene himself, and his result was good by ancient standards, but at least 16% over the true figure, because figures for distances between his chosen locations weren't very accurate.

{* Edit: accurate gnomon measurements, I mean, not accurate distance measurements.}

Second: ancient awareness of the earth's shape was driven by astronomy, not by terrestrial observations. No ancient writer refers to ships' sails going over the horizon (and it's non-trivial to demonstrate the phenomenon unless you have very good vision and exceptionally clear weather). The actual evidence that ancient sources cite for the earth's shape is astronomical, not terrestrial: the spherical geometry of the sky, and the fact that every point on earth corresponds to a point in the sky; the two great circles of the sky, the ecliptic and the equator, and the fact that they're at an angle to one another; the fact that lunar eclipses occur at the same time for everyone but the time of day varies according to east-west separation; and so on. More details in this older response here. Ptolemy and Cleomedes do cite a terrestrial observation as a kind of supplement to the astronomical evidence, namely mountainous islands appearing to rise up as you approach them in a ship. (One source, Pliny, does refer to a bright lantern on a ship's mast dropping below the horizon, which is very different from observing contours; but given that the rest of his evidence is complete garbage, I'm not sure we should even trust that this was a real observation.)

Now, on the subject of controversy. The spherical shape of the earth was mainstream knowledge and usually not controversial in antiquity, as I've outlined in another answer from a few months back. It was mainstream enough for Ovid to use it in the opening paragraph of his epic poem the Metamorphoses; and Pliny tells us that Roman travellers often carried a portable gnomon to measure shadows at midday as a measure of latitude. The main difference between then and now is that ancient writers expressed latitude as a ratio between the length of the gnomon and of its shadow; they didn't convert this into an angular measure using degrees, because you need the inverse tan function to do that, and trigonometry wasn't properly developed until much later.

There was at least one strain of flat-earther thought in the Hellenistic period, namely Epicurean philosophy, but I've never come across evidence that it prompted any kind of scientific debate (and I've looked).

The only concrete disagreements I know of were caused by a form of hardline Christian biblical literalism in the 4th-6th centuries CE. In the New Testament, Hebrews 9.1-14 describes an 'earthly booth' (or 'tabernacle') which represents a 'greater and perfect booth', and on this slender basis some leading Christians interpreted Paul as saying that the cosmos is literally the same shape as the Ark of the Covenant. This wave of literalism was mostly in 4th-5th century Syria: the main figures are Ephrem of Syria, John Chrysostom, Diodorus of Tarsus, Severian of Gabala, and Theodore of Mopsuestia. But there were some flat-earthers elsewhere too: the western rhetorician Lactantius in the 3rd-4th centuries, and Cosmas 'Indicopleustes' in the 6th century, who wrote a substantial treatise on flat-earthism called the Christian topography which survives intact.

This wave of literalism lasted a good while, and some of the Syrians I mentioned went to teach in Constantinople. But it didn't last forever. Already in the 6th century John Philoponus was pointing out that the creation story in the Bible has to be interpreted allegorically because the earth is spherical, and plenty of mediaeval writers before, during, and after the time of the Syrian flat-earthers were absolutely clear about the real shape of the earth -- people like Augustine, Macrobius, Boethius, Bede, Photius, Thomas Aquinas, and so on. I can't say I'm fully acquainted with the entire history of flat-earthism, but I'm not aware of any serious flat-earthism in between the time of Cosmas and the revival of biblical literalism in the modern era.