The equiatomic high-entropy alloy of composition of CrNiCoFeMn with an FCC crystal structure was prepared by either induction melting or additive manufacturing with a selective laser melting (SLM) process, starting from mechanically alloyed powders. The as-produced samples of both kinds were cold worked, and in some cases re-crystallized. Unlike induction melting, there is a second phase, which is made of fine nitride and Cr-rich σ phase precipitates, in the as-produced SLM alloy. Young’s modulus and damping measurements, as a function of temperature in the 300–800 K range, were performed on the specimens that were cold-worked and/or re-crystallized. Young’s modulus values of (140 ± 10) GPa and (90 ± 10) GPa were measured from the resonance frequency of free-clamped bar-shaped samples at 300 K for the induction-melted and SLM samples, respectively. The room temperature values increased to (160 ± 10) GPa and (170 ± 10) GPa for the re-crystallized samples. The damping measurements showed two peaks, which were attributed to dislocation bending and grain-boundary sliding. The peaks were superposed on an increasing temperature background.