We report on the properties of eclipsing binaries (EBs) from the Kepler mission with a newly developed photometric modelling code, which uses the light curve, spectral energy distribution of each binary, and stellar evolution models to infer stellar masses without the need for radial velocity (RV) measurements. We present solutions and posteriors to orbital and stellar parameters for 728 systems, forming the largest homogeneous catalogue of full Kepler binary parameter estimates to date. Using comparisons to published RV measurements, we demonstrate that the inferred properties (e.g. masses) are reliable for well-detached main-sequence (MS) binaries, which make up the majority of our sample. The fidelity of our inferred parameters degrades for a subset of systems not well described by input isochrones, such as short-period binaries that have undergone interactions, or binaries with post-MS components. Additionally, we identify 35 new systems which show evidence of eclipse timing variations, perhaps from apsidal motion due to binary tides or tertiary companions. We plan to subsequently use these models to search for and constrain the presence of circumbinary planets in Kepler EB systems.