Is it possible that Euclid plane can be modelled using countable number of points?

Question

I know that the Pythagorean already showed that the set of all rationals cannot measure all "real" distances, but what if we assume that the number of points and distances in Euclid plane is countable (but not rationals). In this case if we add all the irrational numbers obtained in by geometric figures (such as the hypotenuse of right triangles) we still remain with countable set of measures, provided that we forbid the sum of infinite distances... Now suppose that we do forbid the infinite sum of measures, do we still have to assume that the number of points (or distances) is continuous? Can we found other model (preferably countable), beside the $\mathbb{R}^2$ Cartesian model, that models the Euclidean plane?

Answer 1

A contructible number is a real number that can be obtained through integers, $+$, $-$, $\times$, $\div$, and square roots.

For example, $7+\sqrt{2-\sqrt3}-\sqrt{\sqrt{17}}$ is constructible. A famous theorem says $\sqrt[3]2$ is not constructible.

Let $C$ be the set of constructible numbers. Then $C^2$ should be a good model. It's the set of points obtained with straightedge and compass, where $(0,0)$ and $(0,1)$ are given.