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Emergent nonreciprocity in active matter drives the formation of self-organized states that transcend the behaviors of equilibrium systems. Integrating experiments, theory and simulations, we demonstrate that active solids composed of living starfish embryos spontaneously transition between stable fluctuating and oscillatory steady states. The nonequilibrium steady states arise from two distinct chiral symmetry breaking mechanisms at the microscopic scale: the spinning of individual embryos resulting in a macroscopic odd elastic response, and the precession of their rotation axis, leading to active gyroelasticity. In the oscillatory state, we observe long-wavelength optical vibrational modes that can be excited through mechanical perturbations. Strikingly, these excitable nonreciprocal solids exhibit nonequilibrium work generation without cycling protocols, due to coupled vibrational modes. Our work introduces a novel class of tunable nonequilibrium processes, offering a framework for designing and controlling soft robotic swarms and adaptive active materials, while opening new possibilities for harnessing nonreciprocal interactions in engineered systems.