Chandra X-ray Observatory imaging spectroscopy of the starburst galaxy Henize 2-10 (He 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(-1) for reasonable power law or blackbody models, consistent with accretion onto a >50 M-circle dot 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(-1), 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 similar or equal to 0.2 keV and kT similar or equal to 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 Ha emission, the composition of the X-ray plasma likely reflects mixing of the ambient cool/warm interstellar medium (ISM) with an even hotter, low emission measure plasma, thereby explaining the similar to 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 similar to 30 '', are enveloped within the 8 x 10(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 He 2-10 is in the very early stages of developing a galactic wind.