A bstract Atomic dark matter is a simple but highly theoretically motivated possibility for an interacting dark sector that could constitute some or all of dark matter. We perform a comprehensive study of precision cosmological observables on minimal atomic dark matter, exploring for the first time the full parameter space of dark QED coupling and dark electron and proton masses ( α D , m e D , m p D ) as well as the two cosmological parameters of aDM mass fraction f D and temperature ratio ξ at time of SM recombination. We also show how aDM can accommodate the ( H 0 , S 8 ) tension from late-time measurements, leading to a better fit than ΛCDM or ΛCDM + dark radiation. Furthermore, including late-time measurements leads to closed contours of preferred ξ and dark hydrogen binding energy. The dark proton mass is seemingly unconstrained. Our results serve as an important new jumping-off point for future precision studies of atomic dark matter at non-linear and smaller scales.