We study the atomic physics and the astrophysical implications of a model in which the dark matter is the analog of hydrogen in a secluded sector. The self interactions between dark matter particles include both elastic scatterings as well as inelastic processes due to a hyperfine transition. The self-interaction cross sections are computed by numerically solving the coupled Schr{o}dinger equations for this system. We show that these self interactions exhibit the right velocity dependence to explain the low dark matter density cores seen in small galaxies while being consistent with all constraints from observations of clusters of galaxies. For a viable solution, the dark hydrogen mass has to be in 10--100 GeV range and the dark fine-structure constant has to be larger than 0.02. Precisely for this range of parameters, we show that significant cooling losses may occur due to inelastic excitations to the hyperfine state and subsequent decays, with implications for the evolution of low-mass halos and the early growth of supermassive black holes. Cooling from excitations to higher $n$ levels of dark hydrogen and subsequent decays is possible at the cluster scale, with a strong dependence on halo mass. Finally, we show that the minimum halo mass is in the range of $10^{3.5}$ to $10^7 M_odot$ for the viable regions of parameter space, significantly larger than the typical predictions for weakly-interacting dark matter models. This pattern of observables in cosmological structure formation is unique to this model, making it possible to rule in or rule out hidden sector hydrogen as a viable dark matter model.
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