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(2(1,1) -2(0,2)) 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) x 10(12) L (circle dot) in West and (27 +/- 4) x 10(11) L (circle dot) 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) x 10(11) M (circle dot) in West and (3.1 +/- 2.7) x 10(10) M (circle dot) 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-H-2 conversion factor (alpha (CO)) 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.