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I was reading this, and I am left wondering actually how good classical physics, Newtonian mechanics or general relativity, is in very large length scales, as an approximation?

Therefore, I would like to ask: to what scale, order of magnitude of length, classical mechanics is observationally/ experimentally verified? How is the observation done? $^{\dagger}$

P.S:I am trying to ask a question as well posed as I can, if you find any problems - please at least comment, such that I will learn.

$\dagger:$ I would like an answer that a well prepared physics undergrad can understand. (or at least, a master degree physics student)

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Shing
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The answer is (as always with general questions like this) that it all depends on what physical process you are looking at.

The most fundament description of nature that we know is in terms of quantum mechanics (quantum field theory). However on macroscopic scales classical physics is our working model. For any physical process we can, at least in principle, estimate the corrections due to quantum mechanical effects and these are generally so small we can never hope to measure them. Thus in practice we don't really try to verify classical physics observationally on macroscopic scales, we rather try to observe any quantum mechanical effects whenever they can be observed. Having said that, there are alot of macroscopic physics which we cannot model without taking quantum mechanical effects into account in our classical description (e.g. what happens inside stars).

Leaving that point aside and to give you a concrete example to illustrate how small quantum effects usually are lets take gravity and the Newtonian inverse square law as an example. First of all this law gets corrections from considering the more complete theory of General Relativity and this is often a very small effect. In the solar-system the correction is of the order of $10^{-5}$ and when we try to test gravity these are usually the corrections we try to measure. The quantum corrections are much much smaller. Just from dimensional analysis these can be estimated to be of the order

$$\frac{G\hbar}{r^2c^3} = \left(\frac{r_{\rm Pl}}{r}\right)^2$$

where $r_{\rm Pl} \sim 10^{-35}$m. In the solar-system this is a correction of the order of $10^{-80}$.

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