Hyperuniformity in driven disordered systems [1,2] and giant number fluctuations in active matter [3,4] represent opposite ends in a spectrum of statistical correlations found in physical systems outside of thermal equilibrium. Despite both these phenomena representing milestones in our understanding of non-equilibrium physics -- with potential applications ranging from optimal packing patterns and transport in biological matter [5,6] to the fabrication of disordered materials with photonic bandgaps [7-10] --, the possibility of them being connected has remained so far elusive. Using a combination of experimental and theoretical work on active nematic defects, we demonstrate, through the notion of critical absorbing states [11,12], that such a spectrum is in fact closed, and that active matter can in fact be hyperuniform.