Chandra X-ray imaging spectroscopy of the starburst galaxy Henize 2-10 reveals a strong nuclear point source and at least two fainter compact sources embedded within a more luminous diffuse thermal component. Spectral fits to the nuclear X-ray source imply an unabsorbed X-ray luminosity L_x >10^40 erg/s for reasonable power law or blackbody models, consistent with accretion onto a >50 solar mass black hole behind a foreground absorbing column of N_H>10^23 /cm^2. Two of these point sources have L_x=2-5 x 10^38 erg/s, comparable to luminous X-ray binaries. These compact sources constitute a small fraction (<16%) of the total X-ray flux from He~2-10 in the 0.3--6.0 keV band and just 31% of the X-rays in the hard 1.1--6.0 keV band which is dominated by diffuse emission. Two-temperature solar-composition plasmas (kT~0.2 keV and kT~0.7 keV) fit the diffuse X-ray component as well as single-temperature plasmas with enhanced alpha/Fe ratios. Since the observed radial gradient of the X-ray surface brightness closely follows that of the Halpha emission, the composition of the X-ray plasma likely reflects mixing of the ambient cool/warm ISM with an even hotter, low emission measure plasma, thereby explaining the ~solar ISM composition. Aperture synthesis 21-cm maps show an extended neutral medium to radii of 60 so that the warm and hot phases of the ISM, which extend to ~30, are enveloped within the 8x10^20 /cm^2 contour of the cool neutral medium. This extended neutral halo may serve to inhibit a starburst-driven outflow unless it is predominantly along the line of sight. The high areal density of star formation can also be reconciled with the lack of prominent outflow signatures if Henize 2-10 is in the very early stages of developing a galactic wind.
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