The classical toolkit is two instruments, and the constraints placed on them matter as much as the instruments themselves. The compass draws a circle given a centre and a point on it. The straightedge draws the line through two given points. That is all. In particular the straightedge is unmarked: it carries no scale, so it cannot be used to measure, to transfer a distance, or to slide until two marks land on two given curves.
That last exclusion is the significant one. A construction that slides a marked ruler until two points on it fall on two given curves is called a neusis, and it is genuinely more powerful than compass and straightedge. Archimedes used neusis; so did Nicomedes and Pappus. With a marked ruler an arbitrary angle can be trisected, a cube can be doubled, and the regular heptagon and nonagon can be constructed exactly, all of which the unmarked tools provably cannot do. The Greeks knew these methods and used them, but treated them as a separate class of solution.
So the famous impossibilities of classical geometry are impossibilities relative to a chosen restriction, not limits on geometry itself. Doubling the cube and trisecting the angle become straightforward the moment a mark is allowed on the ruler. Squaring the circle does not, because that one fails for a deeper reason. Paper folding is more powerful still: origami constructions solve cubic equations, and will produce both the heptagon and the trisected angle.
Why the restriction was honoured is a question about Greek mathematical culture rather than about lines. Whatever the reason, it turned out to be the more interesting constraint, because it is the one sharp enough to be settled by proof. The theory that eventually answered these questions is algebraic. Each new point in a compass-and-straightedge construction lies in a field extension of degree one or two over the previous one, so every constructible length has degree a power of 2 over the rationals, and anything requiring an irreducible cubic is out of reach.