By the time of Aristotle in the late 300s, though, there was pretty substantial evidence that the Earth was round and Aristotle collected it all together. Broadly, the arguments could be put into two categories: empirical arguments and theoretical arguments.
The empirical arguments were pretty clever. Astronomers noted that whenever there was a lunar eclipse, the shadow of the Earth was always curved the same way. Since lunar eclipses occurred at different times and different locations on the sky, it could only produce the same shadow if the Earth itself was spherical. Another argument relied on a somewhat fortunate coincidence that one of the brightest stars in the sky, Canopus, is right on the horizon at the latitude of Greece. Astronomers noticed that they could see this star low on the horizon when the traveled to southern cities like Alexandria, but could not see it in northern cities like Athens. This also implied that the Earth was round. Later on Eratosthenes was able to use this principle to measure the size of the Earth and got the answer right to within 1% (although he probably got a bit lucky since he seems to have made two mistakes which partially cancelled each other out).
Aristotle's theoretical justification for a round Earth wasn't too far off the mark, either. His basic argument was that because matter tends to fall to the center of the Earth, if the Earth was some other shape, the matter would rearrange itself until it achieved a spherical shape. His theory of motion was all wrong, but the basic idea that a system with radial forces would produce spheres was correct.
(Shameless plug: I do a podcast on the history of astronomy, so if you want to hear more on the subject I have much more to say [1].)