It is a fact plain to the eye. According to a simple reading of many classic Jewish texts, our tradition is incorrect about science in more than a few instances. The religious Jew generally sees two ways to (actively) approach this problem (as opposed to just saying it doesn't matter).
- The Torah was given to Chazal, but not science. Modern science can be trusted as being authoritative, as Chazal were simply using the tools of their time. If they would be alive today, they would have adjusted their rulings and statements to be in line with modern science. (Generally, this brings many quotes from the Rambam)
- All of the words of Chazal are necessarily true and beyond dispute. If there is a conflict between modern science at what we see in Chazal, science is wrong. (Some add to this that science has a dishonest agenda of proving Torah to be false)
I do not want to get in to the details of this dispute, as there are volumes online about it already due to the debate of the ban on Rabbi Slifkin's books. The point here is that these two points of view seem to take themselves as exclusive. Either the findings of science are true, or the statements of Chazal are true. There is no in between.
But there is a center line in Kabbalah. In the book Pitchei Shearim, by Rav Yitzhak Isaac Chaver (a student of Rav Menachem Mendel of Shklov, who was a student of the GR"A), in the last section on Iggulim v'Yosher, Rav Chaver explains the following:
This world can be understood through two points of view, one of Iggulim, of General Providence, Nature, or alternatively of Yosher, Personal/Particular Providence, Miracles.
As an example, we see a statement in the Gemara that the sun rises in the east, sets in the west, and then goes above the Rakia (the firmament, or sky, if you would) in order to make the trip back to the east side of the sky. Whereas the science shows us that the sun is really underneath us during the night.
The Truth, according to what is brought there, is that both are right. Chazal were talking about one thing, the Gentiles were talking about another. Two different perspectives, both correct. The entire idea from Chazal is spiritual in nature and not talking about the physical world. We see this play through to the first blessing before Shma during Maariv, many ideas which have no basis in observable physical reality.
Now this obviously does not directly resolve all of the problems mentioned in this debate, however it is an approach. We do not write off Chazal, nor do we write off science. It is difficult for the laymen who has access to secular education but not the secrets of Kabbalah, and therefore this might seem hard to accept. We must approach these seeming contradictions with the understanding that not everything we see in Chazal is meant to be understood literally, and we cannot always assume that we understand what we read to its fullest extent.
For application of this idea:
The Earth does revolve around the sun, and the sun does revolve around the center of the galaxy, and we are moving away from the center of the universe. However, the place of the Holy of Holies is most definitely the center of the universe. The former statement refers to the physical, whereas the latter to the spiritual. The spot of Kodesh HaKodashim on Har HaBayit is the central physical reference point for spiritual physics.