I happened upon this paper by Ramanujan in which he gives an approximation for the side length of a square with area nearly equal to that of a given circle. I don't have much experience with constructions of this variety, really only that which most American students do, that being a course in high school geometry. How exactly does one know they've achieved a valid construction? For instance, how did Ramanujan verify this side length is good approximation, was it simply empirical measurement with a ruler? Also, where does one get the intuition for geometric construction? Apart from reading a classic like Euclid's Elements, and a few other famous constructions like that of the lune to build a repertoire of techniques, is it really just trying some constructions and then measuring to see how close you got to the result?
You have asked several questions.
How exactly does one know they've achieved a valid construction?
You do that by writing a proof using Euclid's axioms and previously proved propositions.
For example, Proposition 10 of Book 1 is a construction that bisects a given segment, with a proof that it's correct: https://mathcs.clarku.edu/~djoyce/elements/bookI/propI10.html
For this one
For instance, how did Ramanujan verify this side length is good approximation, was it simply empirical measurement with a ruler?
Well, with a ruler, surely not. This construction finds an approximation. To see why it's a good one you would have to read it carefully. My guess is that it's a Euclidean construction based on a few terms of a series that converges quickly.
(Since $\pi$ is not constructible with Euclidean tools an exact construction is impossible.)
Also, where does one get the intuition for geometric construction?
With practice. Study how to bisect angles, erect perpendiculars, divide segments in given rational proportions, find square roots and mean proportionals (all these are in Euclid). Then you can use those tools to build what you are asked for.