The protein elastin imparts extensibility, elastic recoil, and resilience to tissues including arterial walls, skin, lung alveoli, and the uterus. Elastin and elastin-like peptides are intrinsically disordered hydrophobic proteins that undergo liquid-liquid phase separation upon self-assembly. Despite extensive study for over eighty years, the structure of elastin remains controversial. We use molecular dynamics simulations on a massive scale to elucidate the structural ensemble of aggregated elastin-like peptides. Consistent with the entropic nature of elastic recoil, the aggregated state is stabilized both by the hydrophobic effect and by conformational entropy. The polypeptide backbone forms transient, sparse hydrogen-bonded turns and remains significantly hydrated even as self-assembly triples the extent of nonpolar side-chain contacts. The assembly approaches a maximally-disordered, melt-like state, which may be called the liquid state of proteins. These findings resolve long-standing controversies regarding elastin structure and function and afford insight of broad relevance to the phase separation of disordered proteins.