Generalizations of Mordell's theorem on rational quadrilaterals

Question

Mordell proved that every quadrilateral in the Euclidean plane is arbitrarily close to a quadrilateral whose sides and diagonals are rational. What if we take five points in the plane? Is there a choice of 5 bad points so that for some positive epsilon, whenever the points are moved at most distance epsilon then two of them are at irrational distance?