In a sample of local active galactic nuclei (AGNs) studied at a spatial resolution on the order of 10 pc, we show that the interstellar medium traced by the molecular hydrogen ν = 1-0 S(1) line at 2.1 μm forms a geometrically thick, clumpy disk. The kinematics of the molecular gas reveals general rotation, although an additional significant component of random bulk motion is required by the high local velocity dispersion. The size scale of the typical gas disk is found to have a radius of ∼30 pc with a comparable vertical height. Within this radius, the average gas mass is estimated to be ∼10 7 M⊙ based on a typical gas mass fraction of 10%, which suggests column densities of NH ∼ 5 × 1023 cm-2. Extinction of the stellar continuum within this same region suggests lower column densities of NH ∼ 2 × 1022 cm-2, indicating that the gas distribution on these scales is dominated by dense clumps. In half of the observed Seyfert galaxies, this lower column density is still great enough to obscure the AGN at optical/infrared wavelengths. We conclude, based on the spatial distribution, kinematics, and column densities that the molecular gas observed is spatially mixed with the nuclear stellar population and is likely to be associated with the outer extent of any smaller scale nuclear obscuring structure. Furthermore, we find that the velocity dispersion of the molecular gas is correlated with the star formation rate per unit area, suggesting a link between the two phenomena, and that the gas surface density follows known "Schmidt-Kennicutt" relations. The molecular/dusty structure on these scales may be dynamic since it is possible that the velocity dispersion of the gas, and hence the vertical disk height, is maintained by a short, massive inflow of material into the nuclear region and/or by intense, short-lived nuclear star formation. © 2009. The American Astronomical Society.