  Proteins fold to a specific functional conformation with a densely packed
hydrophobic core that controls their stability. We develop a geometric, yet
all-atom model for proteins that explains the universal core packing fraction
of $\phi_c=0.55$ found in experimental measurements. We show that as the
hydrophobic interactions increase relative to the temperature, a novel jamming
transition occurs when the core packing fraction exceeds $\phi_c$. The model
also recapitulates the global structure of proteins since it can accurately
refold to native-like structures from partially unfolded states.
