If I'm not mistaken it was Aristotle who made the discovery that the earth was, in fact, round; and made the incredibly accurate calculation of the earth's overall size.
I don't know who first determined that it was round, but it was Eratosthenes who made the famous measurement. It's roundness was already known at that point, but I don't know for how long. I suspect that the roundness of the earth has been figured out independently by multiple people, perhaps even in prehistoric times, because there are several clues to it that do not require any special modern equipment to observe. Some that come to mind are:
1. The shadow of the earth on the moon during a lunar eclipse.
2. Ships disappearing below the horizon before
they become too small to see.
3. The curvature of the horizon is actually visible from high places, such as mountaintops, especially if you have a flat landscape to observe, such as an ocean. It's subtle, and the opportunities to see it are few, but it is visible without any special tools. Surely some ancient persons noticed this and pondered what it means.
4. The difference in the position of constellations when viewed from different latitudes. Ancient peoples would have traveled much less, but they likely had much better familiarity with the night sky, so it would almost certainly have been noticed by those few who did make significant north-south treks.