At what distance does magnetic attraction between two properly aligned electrons cancel electric repulsion?

As far as I understand, at no distance does this occur.

For macroscopic permanent dipoles, the magnetic dipole-dipole inteaction falls with distance cubed, so for charged permanent magnetic dipoles, if they are repelling each other at some distance, your only hope is to try to bring them closer. You'll probably get some induced electric polarization effects that might lead to some additional attraction, at some length.

But electrons are not like macroscopic permanent dipoles. From a classical perspective, once electrons stop moving relative to one another, they experience just repulsive force due to Coulomb's law; the spin-related effects aren't strong enough to produce a bound state for isolated electrons. In general, other matter must work to help localize electrons near each other (e.g. as in an atom) or mediate an effective bound state (e.g. phonon-mediated Cooper pairs in superconductors).

However, there is an interesting popular science article and associated journal article by M.N. Chernodub in which it is speculated one can "zap" space with a (extremely, extremely) powerful magnetic field to form Cooper pairs in vacuum. It might be fun to read this, even though it is far out of reach of today's technology.