Weyl semimetals have been intensely studied as a three-dimensional realization of a Dirac-like excitation spectrum where the conduction bands and valence bands touch at isolated Weyl points in momentum space. Like in graphene, this property entails various peculiar electronic properties. However, recent theoretical studies have suggested that resonant scattering from rare regions can give rise to a nonzero density of states even at charge neutrality. Here, we give a detailed account of this effect and demonstrate how the semimetallic nature is suppressed at the lowest scales. To this end, we develop a self-consistent T -matrix approach to investigate the density of states beyond the limit of weak disorder. Our results show a nonvanishing density of states at the Weyl point, which exhibits a nonanalytic dependence on the impurity density. This unusually strong effect of rare regions leads to a revised estimate for the conductivity close to the Weyl point and emphasizes possible deviations from semimetallic behavior in dirty Weyl semimetals at charge neutrality even with very low impurity concentration.