Entanglement is typically created via systematic intervention in the time evolution of an initially unentangled state, which can be achieved by coherent control, carefully tailored nondemolition measurements, or dissipation in the presence of properly engineered reservoirs. In this Letter we show that two-component Fermi gases at ~μK temperatures naturally evolve, in the presence of reactive two-body collisions, into states with highly entangled (Dicke-type) spin wave functions. The entanglement is a steady-state property that emerges-without any intervention-from uncorrelated initial states, and could be used to improve the accuracy of spectroscopy in experiments with fermionic alkaline earth atoms or fermionic ground state molecules.