Based on new and archival Chandra observations of the Sombrero galaxy (M 104 = NGC 4594), we study the X-ray emission from its nucleus and the extended X-ray emission in and around its massive stellar bulge. We find that the 0.3-8 keV luminosity of the nucleus appears constant at 2.4 × 1040 erg s-1, or 10-7 of its Eddington luminosity, on three epochs between 1999 December and 2008 April, but drops by a factor of two in the 2008 November observation. The 2-6 keV unresolved emission from the bulge region closely follows the K-band starlight and most likely arises from unresolved stellar sources. At lower energies, however, the unresolved emission reaches a galactocentric radius of at least 23kpc, significantly beyond the extent of the starlight, clearly indicating the presence of diffuse hot gas. We isolate the emission of the gas by properly accounting for the emission from unresolved stellar sources, predominantly cataclysmic variables and coronally active binaries, whose quasi-universal X-ray emissivity was recently established. We find a gas temperature of 0.6 keV with little variation across the field of view, except for a lower temperature of 0.3 keV along the stellar disk. The metal abundance is not well constrained due to the limited counting statistics, but is consistent with metal enrichment by Type Ia supernovae (SNe Ia). We measure a total intrinsic 0.3-2 keV luminosity of 2 × 10 39 erg s-1, which corresponds to only 1% of the available energy input by SNe Ia in the bulge, but is comparable to the prediction by the latest galaxy formation models for disk galaxies as massive as Sombrero. However, such numerical models do not fully account for internal feedback processes, such as nuclear feedback and stellar feedback, against accretion from the intergalactic medium. On the other hand, we find no evidence for either the nucleus or the very modest star-forming activities in the disk to be a dominant heating source for the diffuse gas. We also show that neither the expected energy released by SNe Ia nor the expected mass returned by evolved stars is recovered by observations. We argue that in Sombrero a galactic-scale subsonic outflow of hot gas continuously removes much of the "missing" energy and mass input from the bulge region. The observed density and temperature distributions of such an outflow, however, continue to pose challenges to theoretical studies.