  The full ``classical" Dirac-Maxwell equations are considered under various
simplifying assumptions. A reduction of the equations is performed in the case
when the Dirac field is {\em static} and a further reduction is performed in
the case of {\em spherical symmetry}. These static spherically symmetric
equations are examined in some detail and a numerical solution presented. Some
surprising results emerge:
  * Spherical symmetry necessitates the existence of a magnetic monopole.
  * There exists a uniquely defined solution, determined only by the demand
that the solution be analytic at infinity.
  * The equations describe highly compact objects with an inner onion-like
shell structure.
