Different proposed EOSs predict a broad range of neutron star radii. The pure hydrogen atmospheres of transiently accreting neutron stars in binaries greatly simplify radiative transfer and make possible accurate radius measurements in these systems. In addition, one can in principle measure both the radius and mass of the neutron star from the broad-band thermal emission alone -- that is, in the absence of any spectral lines. I will give background for what powers the luminosity of these neutron stars, and how the spectrum would be affected by continued active accretion in quiescence. I will review the state of observations -- some objects with spectral uncertainties at the few per cent level, with the total uncertainty dominated by the distance. Finally, I will project long-term expectations of this approach for placing useful constraints on the dense matter EOS.