Measurements of the 21 cm line emission by residual cosmic hydrogen after reionization can be used to trace the power spectrum of density perturbations through a significant fraction of the observable volume of the Universe. We show that a dedicated 21 cm observatory could probe a number of independent modes that is 2 orders of magnitude larger than currently available, and enable a cosmic-variance limited detection of the signature of a neutrino mass approximately 0.05 eV. The evolution of the linear growth factor with redshift could also constrain exotic theories of gravity or dark energy to an unprecedented precision.