Halide double perovskites with alternating silver and pnictogen cations are an emerging family of photoabsorber materials with robust stability and band gaps in the visible range. However, the nature of optical excitations in these systems is not yet well understood, limiting their utility. Here, we use ab initio many-body perturbation theory within the GW approximation and the Bethe-Salpeter equation approach to calculate the electronic structure and optical excitations of the double perovskite series Cs₂AgBX₆, with B = Bi³⁺, Sb³⁺ and X = Br^-, Cl^-. We find that these materials exhibit strongly localized resonant excitons with energies from 170 to 434 meV below the direct band gap. In contrast to lead-based perovskites, the Cs₂AgBX₆ excitons are computed to be non-hydrogenic with anisotropic effective masses and sensitive to local field effects, a consequence of their chemical heterogeneity. Our calculations demonstrate the limitations of the Wannier-Mott and Elliott models for this class of double perovskites and contribute to a detailed atomistic understanding of their light-matter interactions.