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We have undertaken an X-ray survey of a far-infrared flux-limited sample of seven nearby edge-on starburst galaxies. The data are presented in the first paper of this series by Dahlem, Weaver, & Heckman. Here, we examine the two X-ray-brightest sample members, NGC 253 and M82, in a self-consistent manner, taking account of the spatial distribution of the X-ray emission in choosing our spectral models. Both galaxies require at least three components to model the 0.1-10.0 keV spectrum, but the modeling is by no means unique. There is significant X-ray absorption in the disk of NGC 253. When this is accounted for in spectral fits to the emission between ∼0.2 and 2.0 keV, we find that multitemperature thermal plasma models with significant underlying soft X-ray absorption are more consistent with the imaging data than single-temperature models with highly subsolar abundances or models with minimal absorption and nonequilibrium thermal ionization conditions, as have been proposed by others. Our three-component models do not require absolute abundances that are inconsistent with solar values or unusually supersolar ratios of the α-burning elements with respect to Fe (as claimed previously). We conclude that with current data, the technique of measuring abundances in starburst galaxies via X-ray spectral modeling is highly uncertain. Much improved spatial/spectral resolution and increased sensitivity are required. Based on the pointlike nature of much of the X-ray emission in the PSPC hard-band image of NGC 253, we suggest that a significant fraction of the ``extended'' X-ray emission in the 3-10 keV band seen along the disk of the galaxy with ASCA and BeppoSAX (Cappi et al.) is composed of discrete sources in the disk, as opposed to purely diffuse, hot gas. If a significant amount of the hard X-ray emission were due to unresolved point sources with weak Fe Kα emission (e.g., X-ray binaries), this could explain the low Fe abundances of ~(1)/(4) solar derived by Cappi et al. for pure thermal models.