There is mounting evidence that compact elliptical galaxies ( CEGs ) are ‘ ‘ massive relic galaxies ’ ’ that are local analogs of the high-redshift ‘ ‘ red nuggets ’ ’ thought to represent progenitors of today ’ s early-type galaxies ( ETGs ) . We report the discovery of extended X-ray emission from a hot interstellar / intragroup medium in two CEGs , Mrk 1216 and PGC 032873 , using shallow archival \chandra observations . We find that PGC 032873 has an average gas temperature k _ { B } T = 0.67 \pm 0.06 keV within a radius of 15 kpc of the galaxy , and a luminosity L _ { x } = ( 1.8 \pm 0.2 ) \times 10 ^ { 41 } erg s ^ { -1 } within a projected radius of 100 kpc , the latter of which is estimated by extrapolating the fitted \beta model to the surface brightness . For Mrk 1216 , which is closer and more luminous [ L _ { x } ( < 100 ~ { } kpc ) = ( 12.1 \pm 1.9 ) \times 10 ^ { 41 } erg s ^ { -1 } ] , the data are of sufficient quality to perform a spatially resolved spectral analysis in seven circular annuli out to a radius of 73 kpc . Using an entropy-based hydrostatic equilibrium procedure , we obtain good constraints on the H -band stellar mass-to-light ratio , M _ { stars } / L _ { H } = 1.33 \pm 0.21 solar , in good agreement with that obtained from stellar dynamical studies , which supports the hydrostatic equilibrium approximation for this galaxy . We obtain a mass-weighted density slope 2.22 \pm 0.08 within R _ { e } consistent with other CEGs and normal local ETGs , while we find the dark matter ( DM ) fraction within R _ { e } f _ { DM } = 0.20 \pm 0.07 to be similar to local ETGs . We place a constraint on the SMBH mass , M _ { BH } = ( 5 \pm 4 ) \times 10 ^ { 9 } M _ { \odot } , with a 90 % upper limit of M _ { BH } = 1.4 \times 10 ^ { 10 } M _ { \odot } , consistent with a recent stellar dynamical measurement . If we assume the Navarro-Frenk-White ( NFW ) DM scale radius does not lie beyond the current extent of the data , we also obtain interesting constraints on the halo concentration ( c _ { 200 } = 17.5 \pm 6.7 ) and mass [ M _ { 200 } = ( 9.6 \pm 3.7 ) \times 10 ^ { 12 } M _ { \odot } ] . The measured c _ { 200 } exceeds the mean \lcdm value ( \approx 7 ) , consistent with a system that formed earlier than the general halo population . We suggest that these galaxies , which reside in group-scale halos , should be classified as fossil groups .