The results of the analysis of 0.4-10.0 keV ASCA spectral analysis of a sample of low-luminosity active galactic nuclei (AGNs) (LLAGNs; M51, NGC 3147, and NGC 4258), low-ionization nuclear emission-line regions (LINERs; NGC 3079, NGC 3310, NGC 3998, NGC 4579, and NGC 4594), and starburst galaxies (M82, NGC 253, NGC 3628, and NGC 6946) are presented. In spite of the heterogeneous optical classifications of these galaxies, the X-ray spectra are fitted well by a a "canonical" model consisting of an optically thin Raymond-Smith plasma "soft" component with T similar to 7 x 10(6) K and a "hard" component that can be modeled by either a power law with a photon index Gamma similar to 1.7 or a thermal bremsstrahlung with T similar to 6 x 10(7) K. The soft-component absorption is typically less than 10(21) cm(-2), while the hard component is typically absorbed by an additional column on the order of 10(22) cm(-2). The soft-component 0.4-10 keV intrinsic luminosities tend to be on the order 10(39-40) ergs s(-1), while the hard-component luminosities tend to be on the order of 10(40-41) ergs s(-1). The abundances inferred from the fits to the soft component are significantly subsolar. The Fe abundance can be measured independently of the other elemental abundances (dominated by alpha-process elements) in M51, M82, NGC 253, and NGC 4258. In these galaxies the Fe abundance relative to alpha-process elements is also (statistically) significantly subsolar. There is some indication (at a low statistical significance) that the abundance properties of starburst emission from starburst galaxies differs from the starburst emission from low-luminosity AGNs. However, these results on abundances are model dependent. Significant Fe K line emission is observed in M51, M82, NGC 3147, NGC 4258, and NGC 4579. An analysis of the shortterm variability properties was given in Ptak et al. and detailed interpretation of these results will be given in Paper II in this series.