  We numerically study the effect of short ranged potential disorder on
massless noninteracting three-dimensional Dirac and Weyl fermions, with a focus
on the question of the proposed quantum critical point separating the semimetal
and diffusive metal phases. We determine the properties of the eigenstates of
the disordered Dirac Hamiltonian ($H$) and exactly calculate the density of
states (DOS) near zero energy, using a combination of Lanczos on $H^2$ and the
kernel polynomial method on $H$. We establish the existence of two distinct
types of low energy eigenstates contributing to the disordered density of
states in the weak disorder semimetal regime. These are (i) typical eigenstates
that are well described by linearly dispersing perturbatively dressed Dirac
states, and (ii) nonperturbative rare eigenstates that are weakly-dispersive
and quasi-localized in the real space regions with the largest (and rarest)
local random potential. Using twisted boundary conditions, we are able to
systematically find and study these two types of eigenstates. We find that the
Dirac states contribute low energy peaks in the finite-size DOS that arise from
the clean eigenstates which shift and broaden in the presence of disorder. On
the other hand, we establish that the rare quasi-localized eigenstates
contribute a nonzero background DOS which is only weakly energy-dependent near
zero energy and is exponentially small at weak disorder. We find that the
expected semimetal to diffusive metal quantum critical point is converted to an
{\it avoided} quantum criticality that is "rounded out" by nonperturbative
effects, with no signs of any singular behavior in the DOS near the Dirac
energy. We discuss the implications of our results for disordered Dirac and
Weyl semimetals, and reconcile the large body of existing numerical work
showing quantum criticality with the existence of the rare region effects.
