We analyze high-resolution (400pc) 220GHz continuum and CO(2-1) ALMA observations of a representative sample of 23 local (z<0.165) ULIRG systems (34 individual nuclei) as part of the Physics of ULIRGs with MUSE and ALMA (PUMA) project. The deconvolved half-light radii of the 220GHz continuum sources are between <60-350 pc (median 90pc). We associate these regions with the regions emitting the bulk of the infrared luminosity. The good agreement, within a factor of 2, between the 220GHz fluxes and the extrapolation of the infrared gray-body, and the small synchrotron and free-free contributions support this assumption. The cold molecular gas emission sizes, r_CO, are 60-700 pc and are similar in advanced mergers and early interacting systems. On average, r_CO are 2.5 times larger than the continuum. We derive L_IR and cold molecular gas surface densities: log Sigma(L_IR)=11.5-14.3 Lsun/kpc^2 and log Sigma(H2)=2.9-4.2 Msun/pc^2. Assuming that the L_IR is produced by star-formation, this corresponds to median Sigma(SFR)=2500 Msun/yr/kpc^2 which would imply extremely short depletion times, <1-15 Myr, and unphysical SF efficiencies >1 for 70% of the sample. Therefore, this favors the presence of obscured AGN that could dominate the L_IR. We also classify the ULIRG nuclei in two groups: (a) compact nuclei (r<130 pc) with high mid-IR excess emission found in optically classified AGN; and (b) nuclei following a relation with decreasing mid-IR excess for decreasing r. 60% of the interacting nuclei lie in the low end (<130 pc) of this relation, while only 30% of the advanced mergers do so, suggesting that in the early interaction phases the activity occurs in more compact and obscured regions. About two thirds of the nuclei are above the Eddington limit which is consistent with the detection of massive outflows in local ULIRGs and the potential role of radiation pressure in the launching process.
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