We discuss observations of the [C II] 158 mum, [O I] 63 pm, [Si II] 35 mum, [O III] 52,88 mum, and [S III] 33 mum fine-structure transitions toward the central 45'' of the starburst galaxies NGC 253 and NGC 3256. The [C II] and [O I] emission probably originates in photodissociated gas at the surfaces of molecular clouds, although a small (less than or similar to 30%) contribution to the [C II] flux from H II regions cannot be ruled out. The [O III] and [S III] lines originate in H II regions and the [Si II] flux is best explained as originating in H II regions with some contribution from photodissociation regions (PDRs). The gas phase silicon abundance is nearly solar in NGC 253, which we interpret as evidence for grain destruction in the starburst region. We find that the photodissociated atomic gas has densities approximately 10(4) cm-3 and temperatures 200-300 K. About 2% of the gas is in this phase. The thermal gas pressure in the PDRs, P(PDR)/k approximately 1-3 x 10(6) K cm-3, might represent the ''typical'' interstellar gas pressure in starburst systems. The FUV radiation fields illuminating the clouds are 10(3)-10(4) stronger than the local Galactic FUV field and come from the contribution of many closely packed 0 and B stars. For the central 250 pc of NGC 253, we find that the H II gas has an average density n(e) approximately 400 cm-3. This corresponds to a thermal pressure P(H II)/k approximately 7 x 10(6) K cm-3 which is approximately P(PDR)//k, suggesting that the ionized gas is in pressure equilibrium with the photodissociated gas at the surfaces of molecular clouds. The H II gas fills a significant fraction, approximately 0.01-0.3, of the volume between the clouds. The effective temperature of the ionizing stars in NGC 253 is greater than or similar to 34,500 K; 2 x 10(5) 07.5 stars would produce the observed Lyman continuum photon luminosity. The average separation between the stars is approximately 3 pc. Applying the simple model for the interstellar medium in galactic nuclei of Wolfire, Tielens, & Hollenbach (1990), we find the molecular gas in the central regions of NGC 253 and NGC 3256 to be distributed in a large number (5 x 10(3)-5 x 10(5)) of small (0.5-2 pc), dense (approximately 10(4) CM-3) clouds (or alternatively ''thin-flattened'' structures) with volume filling factors 10(-3) 10(-2), very different from the local ISM of the Galaxy. We suggest a self-consistent scenario for the ISM in NGC 253 in which clouds and H II gas are in pressure balance with a supernova-shocked, hot 1-3 x 10(6) K, low-density (approximately 1 cm-3), all pervasive medium. A feedback mechanism may be indicated in which the pressure generated by the supernovae compresses the molecular clouds and triggers further massive star formation. The similarity of ISM parameters deduced for NGC 253, NGC 3256, and M82 (Lord et al. 1993) suggests that the ISM properties are independent of the luminosity of the starburst or the triggering mechanism (presumably a bar in NGC 253, tidal interaction in M82, and a merger in NGC 3256), but are rather endemic to starburst systems. The starburst in NGC 3256 appears to be a scaled-up version of the NGC 253 and M82 starbursts.
