The structure of the hot gas and gravitating matter around M87 is constrained using both X-ray images and spectroscopic observations, particularly of the Fe L lines. The data are previously published results from the four focal plane instruments of the Einstein Observatory. Models in which the gravitating mass has a core radius of ≡25 kpc are favored, suggesting that the material is associated with M87 rather than with the Virgo cluster as a whole. The gas structure is consistent with a radiative accretion flow with a mass accretion rate that varies with radius as M≡ 3(R/10 kpc)2/3 M_sun; yr-1 from ≡1 to 70 kpc. About 1 M_sun; yr-1 flows into the central few arcsec of the galaxy.