One thing that isn’t made, aha, explicit in this excellent article, and that maybe should be, is that considering ruler-and-compasses constructions to be explicit turns out to be exactly the same thing as considering taking square roots to be explicit because (once you have the idea of coordinate geometry, which for some reason no one thought of until Descartes) it pretty easily transpires that the numerical operations you can do on coordinates using ruler-and-compasses constructions are exactly arithmetic plus square roots. (Handwavily, the square roots come from the fact that the equation of a circle involves squaring the coordinates.)
And then (this uses, in some sense, “half” of Galois theory, so it’s easier than the insolubility of the quintic) you can use this to prove that indeed you can’t “duplicate the cube” using ruler and compasses (it means taking cube roots, and it turns out that no quantity of arithmetic-plus-square-roots will let you do that), and that another famous ancient problem—trisecting angles—can’t be done with ruler and compasses for essentially the exact same reason, and (this is a bit more difficult; it was one of Gauss’s first important theorems) that the regular polygons you can construct with ruler and compasses are those whose number of sides is a power of 2 times some number of distinct primes of form 2^2^n+1. So you can do 3 and 5 and 17, but not 9 (two 3s: not allowed) and not 11 (wrong sort of prime).
Of course, for all the reasons daozaich gives here, one shouldn’t care too much about what can be done with ruler and compasses, as such. But these are still really cool theorems.
You’re right, I should have made that clearer, thanks!