  We investigate constraints on the Hubble constant ($H_0$) using Baryon
Acoustic Oscillations (BAO) and baryon density measurements from Big Bang
Nucleosynthesis (BBN). We start by investigating the tension between galaxy BAO
measurements and those using the Lyman-$\alpha$ forest, within a Bayesian
framework. Using the latest results from eBOSS DR14 we find that the
probability of this tension being statistical is $\simeq6.3\%$ assuming flat
$\Lambda$CDM. We measure $H_0 = 67.6\pm1.1$ km s$^{-1}$ Mpc$^{-1}$, with a weak
dependence on the BBN prior used, in agreement with results from Planck Cosmic
Microwave Background (CMB) results and in strong tension with distance ladder
results. Finally, we forecast the future of BAO $+$ BBN measurements of $H_0$,
using the Dark Energy Spectroscopic Instrument (DESI). We find that the choice
of BBN prior will have a significant impact when considering future BAO
measurements from DESI.
