The quotient space of the action of $O(n)$ on $\mathbb{R}^n$ is homeomorphic to $[0,\infty)$
I think it is correct. Basically you are saying: $O(n)$ can take a given point to any other point lying in the same sphere centered at the origin. The quotient is thus made of spheres of all possible radii (like layers of an onion), that is why the quotient is essentially (homeomorphic to) the set of all possible radii, $[0,\infty)$