We present Very Large Array observations of the 33 GHz radio continuum emission from 22 local ultraluminous and luminous infrared (IR) galaxies (U/LIRGs). These observations have spatial (angular) resolutions of 30--720 pc (0.07"-0.67") in a part of the spectrum that is likely to be optically thin. This allows us to estimate the size of the energetically dominant regions. We find half-light radii from 30 pc to 1.7 kpc. The 33 GHz flux density correlates well with the IR emission, and we take

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... se sizes as indicative of the size of the region that produces most of the energy. Combining our 33 GHz sizes with unresolved measurements, we estimate the IR luminosity and star formation rate per area, and the molecular gas surface and volume densities. These quantities span a wide range (4 dex) and include some of the highest values measured for any galaxy (e.g., Σ_SFR^33GHz≤ 10^4.1 M_ yr^-1 kpc^-2). At least 13 sources appear Compton thick (N_H^33GHz≥ 10^24 cm^-2). Consistent with previous work, contrasting these data with observations of normal disk galaxies suggests a nonlinear and likely multi-valued relation between SFR and molecular gas surface density, though this result depends on the adopted CO-to-H_2 conversion factor and the assumption that our 33 GHz sizes apply to the gas. 11 sources appear to exceed the luminosity surface density predicted for starbursts supported by radiation pressure and supernovae feedback, however we note the need for more detailed observations of the inner disk structure. U/LIRGs with higher surface brightness exhibit stronger [ Cii] 158μm deficits, consistent with the suggestion that high energy densities drive this phenomenon.