We demonstrate that the complex spatiotemporal structure in active fluids can feature characteristics of hyperuniformity. Using a hydrodynamic model, we show that the transition from hyperuniformity to nonhyperuniformity and antihyperuniformity depends on the strength of active forcing and can be related to features of active turbulence without and with scaling characteristics of inertial turbulence. Combined with identified signatures of Levy walks and nonuniversal diffusion in these systems, this allows for a biological interpretation and the speculation of nonequilibrium hyperuniform states in active fluids as optimal states with respect to robustness and strategies of evasion and foraging.