We examine the impact of baryon acoustic oscillation (BAO) scale measurements on the discrepancy between the value of the Hubble constant ($H_0$) inferred from the local distance ladder and from Planck cosmic microwave background (CMB) data. While the BAO data alone cannot constrain $H_0$, we show that combining the latest BAO results with WMAP, Atacama Cosmology Telescope (ACT), or South Pole Telescope (SPT) CMB data produces values of $H_0$ that are $2.4-3.1sigma$ lower than the distance ladder, independent of Planck, and that this downward pull was less apparent in some earlier analyses that used only angle-averaged BAO scale constraints rather than full anisotropic information. At the same time, the combination of BAO and CMB data also disfavors the lower values of $H_0$ preferred by the Planck high-multipole temperature power spectrum. Combining galaxy and Lyman-$alpha$ forest (Ly$alpha$) BAO with a precise estimate of the primordial deuterium abundance produces $H_0=66.98pm1.18$ km s$^{-1}$ Mpc$^{-1}$ for the flat $Lambda$CDM model. This value is completely independent of CMB anisotropy constraints and is $3.0sigma$ lower than the latest distance ladder constraint, although $2.4sigma$ tension also exists between the galaxy BAO and Ly$alpha$ BAO. These results show that it is not possible to explain the $H_0$ disagreement solely with a systematic error specific to the Planck data. The fact that tensions remain even after the removal of any single data set makes this intriguing puzzle all the more challenging to resolve.
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