You can even find infinitely many such points on the unit circle:  Let $\mathscr S$ be the set of all points on the unit circle such that $\tan \left(\frac {\theta}4\right)\in \mathbb Q$.  If $(\cos(\alpha),\sin(\alpha))$ and $(\cos(\beta),\sin(\beta))$ are two points on the circle then a little geometry tells us that the distance between them is (the absolute value of) $$2 \sin \left(\frac {\alpha}2\right)\cos \left(\frac {\beta}2\right)-2 \sin \left(\frac {\beta}2\right)\cos \left(\frac {\alpha}2\right)$$  and, if the points are both in $\mathscr S$ then this is rational.

Details:  The distance formula is an immediate consequence of the fact that, if two points on the circle have an angle $\phi$ between them, then the distance between them is (the absolute value of) $2\sin \frac {\phi}2$.  For the rationality note that  $$z=\tan \frac {\phi}2 \implies \cos \phi= \frac {1-z^2}{1+z^2} \quad \& \quad  \sin \phi= \frac {2z}{1+z^2}$$



Note:  Of course $\mathscr S$ is dense on the circle.  So far as I am aware, it is unknown whether you can find such a set which is dense on the entire plane.