Is there a name for this "euclidean" set of real numbers? Let $E$ be the smallest set of real numbers such that: (1) $1\in E$, (2) $x \in E \implies x/n\in E$ $\;$($n=1,2, \;...\;$), (3) $x,y \in E \implies |x-y| \in E$, (4) $x,y \in E \implies \sqrt{x^2+y^2} \in E$. The idea is that $E$ is the set of distances generated by euclidean straight-edge and compass constructions from a unit segment.

The set of distances generated by Euclidean straight-edge and compass constructions from a unit segment is the same as the set of coordinates of constructible points. This is called the set of _constructible numbers_ , as you expect.