From the bottom of a 6-foot diameter well, at a depth of say 50 feet, the sky would subtend the same angle to the eye as a 1.5 inch diameter tube of length 1 foot held up to the eye. This is about the same size as the paper tube in American ' kitchen towel' paper. If you do this experiment in the daytime you will see that no stars come out by sighting through such a tube. So, by direct experiment with a similar geometry, the answer is no, you cannot see stars in the daytime at the bottom of a well.
As a caviat, Ken Tapping an astronomer at the NRC Herzberg Institute of Astrophysics notes: "If we were on the Moon's surface, where there is no atmosphere, we could simply shade out the Sun and the reflected glare from the ground and see the stars perfectly well. On the Earth, our atmosphere tends to scatter the sunlight, which is what makes the sky look blue. This blue is sufficiently bright that it is very difficult to see the stars through it, although on really clear days, in dark places such as the bottom of a well, where reflected light from things on the ground isn't reaching your eyes, it is sometimes possible to see a star or two."
Another answer, and far more complete, is given by David Hughes in the Quarterly Journal of the Royal Astron. Soc., 1983, vol. 24, pp 246-257.
This mistaken notion was first mentioned by Aristotle and other ancient sources, and was widely assumed to be correct by many literary sources of the 19th century, and even believed by some astronomers. But every astronomer who has ever tested this by experiment came away convinced it was impossible.
Separate experiments to attempt to see Vega and Pollux through tall chimneys were performed by J. A. Hynek and A. N. Winsor. They were unable to detect the stars under near perfect conditions, even withbinoculars. The daytime sky is simply too bright to allow us to see even the brightest stars (although Sirius can sometimes be glimpsed just after the Sun rises if you know exactly where to look.) Venus can be seen as a tiny white speck but again, you have to be looking exactly at the right spot.
The most likely explanation for the old legend is that stray bits of rubbish get caught in the updraft and catch the sunlight as they emerge from the chimney. It is possible to see stars in the daytime with a good telescope, as long as it has been prefocused and can be accurately pointed at a target.
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