Stress granules (SG) are membrane‐less compartments involved in regulating mRNAs during stress. Aberrant forms of SGs have been implicated in age‐related diseases, such as amyotrophic lateral sclerosis (ALS), but the molecular events triggering their formation are still unknown. Here, we find that misfolded proteins, such as ALS‐linked variants of SOD1, specifically accumulate and aggregate within SGs in human cells. This decreases the dynamics of SGs, changes SG composition, and triggers an aberrant liquid‐to‐solid transition of in vitro reconstituted compartments. We show that chaperone recruitment prevents the formation of aberrant SGs and promotes SG disassembly when the stress subsides. Moreover, we identify a backup system for SG clearance, which involves transport of aberrant SGs to the aggresome and their degradation by autophagy. Thus, cells employ a system of SG quality control to prevent accumulation of misfolded proteins and maintain the dynamic state of SGs, which may have relevance for ALS and related diseases.