We theoretically analyze a novel evaporative cooling scheme applicable to Rydberg chains. We consider an initially thermal chain of circular Rydberg atoms confined to a one-dimensional (1D) setting in the absence of periodic potentials. We show that the evaporation of about one half of the atoms of the chain brings it near its quantum ground state, which is a 1D Rydberg crystal. We describe the scheme thermodynamically, applying the truncated Boltzmann distribution to the collective excitations of the chain, and show that it leads to a novel quasi--equilibrium many-body state. The evaporation is driven by slowly compressing the chain. For a given atom number, this process is adiabatic down to a minimal chain size. At this point, a single atom is expelled, causing the energy and entropy per particle to decrease. Then, adiabatic compression resumes. For longer chains, comprising about $1000$ atoms, we emphasize the quasi-universality of the evaporation curve.