Failure of amorphous solids is fundamental to various phenomena, including landslides and earthquakes. Recent experiments indicate that highly plastic regions form elongated structures that are especially apparent near the maximal shear stress Sigma(max) where failure occurs. This observation suggested that Sigma(max) acts as a critical point where the length scale of those structures diverges, possibly causing macroscopic transient shear bands. Here, we argue instead that the entire solid phase (Sigma < Sigma(max)) is critical, that plasticity always involves system-spanning events, and that their magnitude diverges at Sigma(max) independently of the presence of shear bands. We relate the statistics and fractal properties of these rearrangements to an exponent theta that captures the stability of the material, which is observed to vary continuously with stress, and we confirm our predictions in elastoplastic models.