We theoretically study the single-particle Green function of a three-dimensional disordered Weyl semimetal using a combination of techniques. These include analytic T-matrix and renormalization group methods with complementary regimes of validity and an exact numerical approach based on the kernel polynomial technique. We show that at any nonzero disorder, Weyl excitations are not ballistic: They instead have a nonzero linewidth that for weak short-range disorder arises from nonperturbative resonant impurity scattering. Perturbative approaches find a quantum critical point between a semimetal and a metal at a finite disorder strength, but this transition is avoided due to nonperturbative effects. At moderate disorder strength and intermediate energies the avoided quantum critical point renormalizes the scaling of single-particle properties. In this regime we compute numerically the anomalous dimension of the fermion field and find eta = 0.13 +/- 0.04, which agrees well with a renormalization group analysis (eta = 0.125). Our predictions can be directly tested by ARPES and STM measurements in samples dominated by neutral impurities.