We present accretion disk structure measurements from continuum lags in the Sloan Digital Sky Survey Reverberation Mapping (SDSS-RM) project. Lags are measured using the JAVELIN software from the first-year SDSS-RM g and i photometry, resulting in well-defined lags for 95 quasars, 33 of which have lag S/N > 2 . We also estimate lags using the ICCF software and find consistent results, though with larger uncertainties. Accretion disk structure is fit using a Markov chain Monte Carlo approach, parameterizing the measured continuum lags as a function of disk size normalization, wavelength, black hole mass, and luminosity. In contrast with previous observations, our best-fit disk sizes and color profiles are consistent (within 1.5 ) with the Shakura & Sunyaev analytic solution. We also find that more massive quasars have larger accretion disks, similarly consistent with the analytic accretion disk model. The data are inconclusive on a correlation between disk size and continuum luminosity, with results that are consistent with both no correlation and the Shakura & Sunyaev expectation. The continuum lag fits have a large excess dispersion, indicating that our measured lag errors are underestimated and/or our best-fit model may be missing the effects of orientation, spin, and/or radiative efficiency. We demonstrate that fitting disk parameters using only the highest-S/N lag measurements biases best-fit disk sizes to be larger than the disk sizes recovered using a Bayesian approach on the full sample of well-defined lags.