  The Hubble tension and attempts to resolve it by modifying the physics of (or
at) recombination motivate finding ways to determine $H_0$ and the sound
horizon at the epoch of baryon decoupling $r_{\rm d}$ in ways that neither rely
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 $r_{\rm d}$ and $H_0$ as independent free parameters. It is well known
that BAO gives exquisite constraints on the product $r_{\rm d}H_0$. We show
here that imposing a moderate prior on $\Omega_{\rm m} h^2$ breaks the
degeneracy between $r_{\rm d}$ and $H_0$. Using the latest BAO data, including
the recently released eBOSS DR16, along with a $\Omega_{\rm m} h^2$ prior based
on the Planck best fit $\Lambda$CDM model, we find $r_{\rm d} =143.7 \pm 2.7$
Mpc and $H_0 = 69.6 \pm 1.8$ km/s/Mpc. BAO data therefore prefers somewhat
lower $r_{\rm d}$ and higher $H_0$ than those inferred from Planck data in a
$\Lambda$CDM model. We find similar values when combing BAO with the Pantheon
supernovae, DES galaxy weak lensing, Planck or SPTPol CMB lensing and the
cosmic chronometers data. We perform a forecast for DESI and find that, when
aided with a moderate prior on $\Omega_{\rm m} h^2$, DESI will measure $r_{\rm
d}$ and $H_0$ without assuming a recombination model with an accuracy
surpassing the current best estimates from Planck.
