Eratosthenes' stick experiment can not only tell us about the size of the earth, but can also be used to compute the distance to the sun as well.
In his experiment Eratosthenes assumes that the earth is a globe and that the sun is very far away in his computations for the size of the earth and the distance to the sun. However, if we use his data with the assumption that the earth is flat we can come up with a wildly different calculation for the distance of the sun, showing it to be close to the earth. The sun changes its distance depending on the model of the earth we assume for the experiment.
Millersville University goes over the two ways of interpreting Eratosthenes' data. The first part of the article goes over the interpretation of his data under a Round Earth model, and the bottom part of the article goes over an interpretation of the data under a Flat Earth model.
An Alternative Model
Eratosthenes' model depends on the assumption that the earth is a globe and that the sun is far away and therefore produces parallel rays of light all over the earth. If the sun is nearby, then shadows will change length even for a flat earth. A flat earth model is sketched below. The vertical stick casts shadows that grow longer as the stick moves to the left, away from the closest point to the sun. (The sun is at height h above the earth.)
A little trigonometry shows that
Using the values 50 degrees and 60 degrees as measured on the trip, with b=1000 miles, we find that h is approximately 2000 miles. This relatively close sun would have been quite plausible to the ancients.
Continuing the calculation, we find that a is approximately 2400 miles and the two distances R1 and R2 are approximately 3000 and 3900 miles, respectively.
That is, as we move from Florida to Pennsylvania, our distance from the sun increases by about 30%. As a consequence the apparant size of the sun should decrease by 30%. We see no noticeable change in the apparent size of the sun as we make the trip. We conclude that the flat earth/near sun model does not work.
Above, the author at Millersville University describes how the distance to the sun could be calculated with Eratosthenes's data under the interpretation a Flat Earth. The author agrees that Eratosthenes assumes which world shape is correct with making his calculation. Eratosthenes' calculations are not proof of any particular model, but based on an assumption of which earth model is correct.
Under the assumption of a Flat Earth the sun is computed to be relatively close to the earth's surface, as seen in the above quote.
Under the assumptions of a Round Earth, on the other hand, the sun can be computed to be millions of miles distant.
Erathonese's calculations for the size of the earth and the distance to the sun do not constitute proof in themselves since the values change depending on the shape of the earth we assume. The calculations are only valid if the assumed shape of the earth is true.
Finally, at the bottom of the article the author goes on to mention that a Flat Earth model would not work because the sun's visible diameter must shrink as it recedes. This is correct. Any receding body should shrink in diameter as it recedes into the far distance. However, it should be mentioned that the sun is not just any body, but one of incredible luminosity which catches onto the atmosphere and magnifies the sun's image the further it recedes from the observer.
This is how the sun maintains its diameter throughout the day and is discussed in depth in The Cosmos section of the Wiki: