The Hubble tension and attempts to resolve it by modifying the physics of (or at) recombination motivate finding ways to determine H0 and the sound horizon at the epoch of baryon decoupling rd in ways that rely neither on a recombination model nor on late-time Hubble data. In this work, we investigate what one can learn from the current and future BAO data when treating rd and H0 as independent free parameters. It is well known that baryon acoustic oscillations (BAOs) give exquisite constraints on the product rdH0. We show here that imposing a moderate prior on mh2 breaks the degeneracy between rd and H0. Using the latest BAO data, including the recently released the extended Baryon Oscillation Spectroscopic Survey Data Release 16, along with a mh2 prior based on the Planck best-fit Λ cold dark matter (ΛCDM) model, we find rd = 143.7 2.7 Mpc and H0 = 69.6 1.8 km s−1 Mpc−1. BAO data prefers somewhat lower rd and higher H0 than those inferred from Planck data in a ΛCDM model. We find similar values when combing BAO with the Pantheon supernovae, the Dark Energy Survey Year 1 galaxy weak lensing, Planck or SPTPol cosmic microwave background lensing, and the cosmic chronometer data. We perform a forecast for the Dark Energy Spectroscopic Instrument (DESI) and find that, when aided with a moderate prior on mh2, DESI will measure rd and H0 without assuming a recombination model with an accuracy surpassing the current best estimates from Planck.