The fixed stars can be demonstrated to be very far away, by diurnal parallax measurements (a technique known at least as early as Ancient Greece). Unlike the Sun, Moon, and planets, they do not change position with respect to one another (at least not perceptibly over the span of a human lifetime); the shapes of the constellations are always the same. This makes them a convenient reference background for determining the shape of the Earth. Adding distance measurements on the ground allows calculation of the Earth's size.
The fact that different stars are visible from different locations on the Earth was noticed in ancient times. Aristotle wrote that some stars are visible from Egypt which are not visible from Europe. This would not be possible if the Earth was flat.
At the North Pole it is continuously nighttime for six months of the year and the same hemisphere of stars (a 180° view) are always visible making one counterclockwise rotation every 24 hours. The star Polaris (the "North Star") is almost at the center of this rotation (which is directly overhead). Some of the 88 modern constellations visible are Ursa Major (including the Big Dipper), Cassiopeia, and Andromeda. The other six months of the year, it is continuously daytime and the light from the Sun mostly blots out the stars. (The location of the poles can be defined by these phenomena, which only occur there; more than 24 hours of continuous daylight can occur north of the Arctic Circle and south of the Antarctic Circle.)
At the South Pole, a completely non-overlapping set of constellations are visible during the six months of continuous nighttime, including Orion, Crux, and Centaurus. This 180° hemisphere of stars rotate clockwise once every 24 hours, around a point directly overhead (where there do not happen to be any particularly bright stars).
The fact that the stars visible from the north and south poles do not overlap must mean that the two observation spots are on opposite sides of the Earth, which is not possible if the Earth is a single-sided disc, but is possible for other shapes (like a sphere, but also any other convex shape like a donut or dumbbell).
From any point on the equator, 360° of stars are visible over the course of the night, as the sky rotates around a line drawn from due north to due south (which could be defined as "the directions to walk to get to the poles in the shortest amount of time"). When facing east, the stars visible from the north pole are on the left, and the stars visible from the south pole are on the right. This means the equator must be facing at a 90° angle from the poles.
The direction any intermediate spot on the Earth is facing can also be calculated by measuring the angles of the fixed stars and determining how much of the sky is visible. For example, New York City is about 40° north of the equator. The apparent motion of the Sun blots out slightly different parts of the sky from day to day, but over the course of the entire year it sees a dome of 280° (360° - 80°). So for example, both Orion and the Big Dipper are visible during at least part of the year.
Making stellar observations from a representative set of points across the Earth, combined with knowing the shortest on-the-ground distance between any two given points makes an approximate sphere the only possible shape for the Earth.
Knowing the difference in angle between two points on the Earth's surface and the surface distance between them allows a calculation of the Earth's size. Using observations at Rhodes (in Greece) and Alexandria (in Egypt) and the distance between them, the Ancient Greek philosopher Posidonius actually did use this technique to calculate the circumference of the planet to within perhaps 4% of the correct value (though modern equivalents of his units of measure are not precisely known).