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.