SPT0311-58 is the most massive infrared luminous system discovered so far during the Epoch of Reionization (EoR). In this paper, we present a detailed analysis of the molecular interstellar medium at z = 6.9, through high resolution observations of the CO(6-5), CO(7-6), CO(10-9), [C I](2-1), and p-H2O(21,1 -20,2) lines and dust continuum emissions with the Atacama Large Millimeter/submillimeter Array. The system consists of a pair of intensely star-forming, gravitationally lensed galaxies (labeled West and East). The intrinsic far-infrared luminosity is (16 ± 4) × 1012 Le in West and (27 ± 4) × 1011 Le in East. We model the dust, CO, and [C I] using non-local thermodynamic equilibrium radiative transfer models and estimate the intrinsic gas mass to be (5.4 ± 3.4) × 1011Me in West and (3.1 ± 2.7) × 1010Me in East. We find that the CO spectral line energy distribution in West and East are typical of high-redshift submillimeter galaxies (SMGs). The CO-to-H2 conversion factor (αCO) and the gas depletion timescales estimated from the model are consistent with the high-redshift SMGs in the literature within the uncertainties. We find no evidence of evolution of depletion time with redshift in SMGs at z > 3. This is the most detailed study of molecular gas content of a galaxy in the EoR to date, with the most distant detection of H2O in a galaxy without any evidence for active galactic nuclei in the literature.