Compartmentalization of cellular material in droplet-like structures is a hallmark of liquid-liquid phase separation1,2, but the mechanisms of droplet removal are poorly understood. Evidence suggests that droplets can be degraded by autophagy3,4, a highly conserved degradation system in which membrane sheets bend to isolate portions of the cytoplasm within double-membrane autophagosomes5-7. Here we examine how autophagosomes sequester droplets that contain the protein p62 (also known as SQSTM1) in living cells, and demonstrate that double-membrane, autophagosome-like vesicles form at the surface of protein-free droplets in vitro through partial wetting. A minimal physical model shows that droplet surface tension supports the formation of membrane sheets. The model also predicts that bending sheets either divide droplets for piecemeal sequestration or sequester entire droplets. We find that autophagosomal sequestration is robust to variations in the droplet-sheet adhesion strength. However, the two sides of partially wetted sheets are exposed to different environments, which can determine the bending direction of autophagosomal sheets. Our discovery of this interplay between the material properties of droplets and membrane sheets enables us to elucidate the mechanisms that underpin droplet autophagy, or 'fluidophagy'. Furthermore, we uncover a switching mechanism that allows droplets to act as liquid assembly platforms for cytosol-degrading autophagosomes8 or as specific autophagy substrates9-11. We propose that droplet-mediated autophagy represents a previously undescribed class of processes that are driven by elastocapillarity, highlighting the importance of wetting in cytosolic organization.