We have studied the nuclear activity in a sample of six quiescent early-type galaxies, with new Chandra data and archival HST optical images. Their nuclear sources have X-ray luminosities ~1038-1039 ergs s-1 (LX/LEdd ~ 10-8 to 10-7) and colors or spectra consistent with accreting supermassive black holes (SMBHs), except for the nucleus of NGC 4486B, which is softer than typical AGN spectra. In a few cases, the X-ray morphology of the nuclear sources shows hints of marginally extended structures, in addition to the surrounding diffuse thermal emission from hot gas, which is detectable on scales 1 kpc. In one case (NGC 5845), a dusty disk may partially obstruct our direct view of the SMBH. We have estimated the temperature and density of the hot interstellar medium, which is one major source of fuel for the accreting SMBH; typical central densities are ne ≈ (0.02 ± 0.01) cm-3. Assuming that the hot gas is captured by the SMBH at the Bondi rate, we show that the observed X-ray luminosities are too faint to be consistent with standard disk accretion, but brighter than predicted by radiatively inefficient solutions (e.g., advection-dominated accretion flows [ADAFs]). In total, there are ≈20 galaxies for which SMBH mass, hot gas density, and nuclear X-ray luminosity are simultaneously known. In some cases, the nuclear sources are brighter than predicted by the ADAF model; in other cases, they are consistent or fainter. We discuss the apparent lack of correlations between Bondi rate and X-ray luminosity and suggest that, in order to understand the observed distribution, we need to know two additional parameters: the amount of gas supplied by the stellar population inside the accretion radius, and the fraction (possibly 1) of the total gas available that is accreted by the SMBH. We leave a detailed study of these issues to a subsequent paper.