replicate-iterate-innovate
"Or the sun is regular-size, but the stars are incredibly far away; so far away that their precession takes aeons, and all we observe are the brightest and closest. I could probably do some trigonometry to check, one way or the other - really, anyone with some straight rods and standardized units of measurement ought to be able to replicate Eratosthenes' experiment, it's just trigonometry plus facts of geography.
"It goes like this: The curvature of the planet is such that there is some point, on some day, where the sun will not cast a visible shadow from an upright pole.
"You find that point, and then take another measurement of the shadow cast by a pole of the same height, a reasonably large distance away to the north or south, and preferably at the same elevation. You should observe, if my model of the world is at all accurate, that there is a shadow cast.
"This allows you to calculate the circumference of the earth reasonably well, despite the fact that it's not quite right since the planet's spinning squishes it a bit and, uh, elevation is a factor.
"Then, using those numbers, you can construct a triangle with those rods as two points and the sun as another; the rest is just plugging it into a squared plus b squared minus two a b cosine C-the-angle, slash otherwise exploiting the fact that triangles are pretty easy to pin down if you know enough about them - and I'm pretty sure we would, in this case.
"But actually...have you ever tested that 'the sun is less magical than other stars' hypothesis? It seems eminently measureable."