This actually is easier to work out than you might think, and can be done with fairly rudimentary techniques. Really, all you need is two people at some known (fairly large) distance apart from one another with sticks and rulers. In what follows, I'll outline a method for calculating the circumference of the earth due to Eratosethenes (276-194 BC).

Say you live in Athens and have a friend in Alexandria. You and your friend decide to run a little experiment. Your friend has told you that at a certain time of day in Alexandria, a stick standing straight up (on flat ground in direct sunlight) casts no shadow. From this, you reason that the sun must be directly above Alexandria at that time. At that same time that no shadow is cast in Alexandria, you measure the angle and length of a shadow being cast in Athens.

This may not seem like much, but it actually gives you everything you need to know to (i) infer that the Earth is a sphere and (ii) make a surprisingly accurate estimate of the circumference of the Earth. I'll spare you the gory mathematical details, but the angles you will measure in Athens show you that the surface of the Earth is curved, since if the Earth's surface were flat the angle and length of the shadow would be different from what we actually observe. From there, you can use some geometry and the distance between the two cities to work out the circumference of the Earth.