We experimentally study the emergence of microcanonical equilibrium states in the turbulent relaxation dynamics of a two-dimensional chiral vortex gas. Same-sign vortices are injected into a quasi-two-dimensional disk-shaped atomic Bose-Einstein condensate using a range of mechanical stirring protocols. The resulting long-time vortex distributions are found to be in excellent agreement with the mean-field Poisson Boltzmann equation for the system describing the microcanonical ensemble at fixed energy H and angular momentum M. The equilibrium states are characterized by the corresponding thermodynamic variables of inverse temperature β̂ and rotation frequency .̂ We are able to realize equilibria spanning the full phase diagram of the vortex gas, including on-axis states near zero temperature, infinite temperature, and negative absolute temperatures. At sufficiently high energies, the system exhibits a symmetry-breaking transition, resulting in an off-axis equilibrium phase at negative absolute temperature that no longer shares the symmetry of the container. We introduce a point-vortex model with phenomenological damping and noise that is able to quantitatively reproduce the equilibration dynamics.