The question may seem a bit stupid at a first glance, but let me explain.
A constructible number is constructed by a sequence of compass and straightedge constructions that always begin with two points $0$ and $1$.
For example, it is usually proved that squaring the circle by compass and straightedge is impossible because $\sqrt{\pi}$ is not a constructible number. While I understand that given two points $0,1$, one cannot construct the number $\sqrt{\pi}$ by compass and straightedge, I don't understand why given a circle, one cannot construct a square with the circle's area by compass and straightedge.
The "starting configurations" are given "in intrinsic form", without reference to a particular coordinate system and it is not always possible to choose a coordinate system such that the starting configurations are definable from points with constructible coordinates.
So why do non-constructible numbers imply the impossibility of geometric constructions?