  We consider the massless Dirac operator with the MIT bag boundary conditions
on an unbounded three-dimensional circular cone. For convex cones, we prove
that this operator is self-adjoint defined on four-component $H^1$--functions
satisfying the MIT bag boundary conditions. The proof of this result relies on
separation of variables and spectral estimates for one-dimensional fiber
Dirac-type operators. Furthermore, we provide a numerical evidence for the
self-adjointness on the same domain also for non-convex cones. Moreover, we
prove a Hardy-type inequality for such a Dirac operator on convex cones, which,
in particular, yields stability of self-adjointness under perturbations by a
class of unbounded potentials. Further extensions of our results to Dirac
operators with quantum dot boundary conditions are also discussed.
