Using the PHANGS-ALMA CO(2-1) survey, we characterize molecular gas properties on similar to 100 pc scales across 102,778 independent sightlines in 70 nearby galaxies. This yields the best synthetic view of molecular gas properties on cloud scales across the local star-forming galaxy population obtained to date. Consistent with previous studies, we observe a wide range of molecular gas surface densities (3.4 dex), velocity dispersions (1.7 dex), and turbulent pressures (6.5 dex) across the galaxies in our sample. Under simplifying assumptions about subresolution gas structure, the inferred virial parameters suggest that the kinetic energy of the molecular gas typically exceeds its self-gravitational binding energy at similar to 100 pc scales by a modest factor (1.3 on average). We find that the cloud-scale surface density, velocity dispersion, and turbulent pressure (1) increase toward the inner parts of galaxies, (2) are exceptionally high in the centers of barred galaxies (where the gas also appears less gravitationally bound), and (3) are moderately higher in spiral arms than in inter-arm regions. The galaxy-wide averages of these gas properties also correlate with the integrated stellar mass, star formation rate, and offset from the star-forming main sequence of the host galaxies. These correlations persist even when we exclude regions with extraordinary gas properties in galaxy centers, which contribute significantly to the inter-galaxy variations. Our results provide key empirical constraints on the physical link between molecular cloud populations and their galactic environment.