  It is shown how point charges and point dipoles with finite self-energies can
be accomodated into classical electrodynamics. The key idea is the introduction
of constitutive relations for the electromagnetic vacuum, which actually
mirrors the physical reality of vacuum polarization. Our results reduce to
conventional electrodynamics for scales large compared to the classical
electron radius $r_0\approx 2.8\times10^{-13}$ cm. A classical simulation for a
structureless electron is proposed, with the appropriate values of mass, spin
and magnetic moment.
