We present a detailed case study of the diffuse X-ray and H-alpha emission in the halo of NGC 253, a nearby edge-on starburst galaxy driving a galactic superwind. The arcsecond spatial resolution of the Chandra ACIS instrument allows us to study the spatial and spectral properties of the diffuse X-ray emitting plasma with greatly superior spatial and spectral resolution compared to previous X-ray instruments. We find statistically significant structure within the X-ray diffuse emission on angular scales down to ~10 arcsec (~130 pc). There is no statistically significant evidence for any spatial variation in the spectral properties of the diffuse emission, over scales from ~400 pc to ~3 kpc. We show that the X-shaped soft X-ray morphology of the superwind, previously revealed by ROSAT, is matched by very similar X-shaped H-alpha emission, extending at least 8 kpc above the plane of the galaxy. In the northern halo the X-ray emission appears to lie slightly interior to the boundary marked by the H-alpha emission. The total 0.3-2.0 keV energy band X-ray luminosity of the northern halo, L_X ~ 5e38 erg/s, is very similar to the halo H-alpha luminosity of L_Ha ~ 4e38 erg/s, both of which are a small fraction of the estimated wind energy injection rate of ~1e42 erg/s from supernovae in the starburst. We show that there are a variety of models that can simultaneously explain spatially-correlated X-ray and H-alpha emission in the halos of starburst galaxies. These findings indicate that the physical origin of the X-ray-emitting million-degree plasma in superwinds is closely linked to the presence of much cooler and denser T ~ 1e4 K gas, not only within the central kpc regions of starbursts, but also on ~10 kpc-scales within the halos of these galaxies. (Abridged)
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