We present a detailed case study of the diffuse X-ray and Hα emission in the halo of NGC 253, a nearby edge-on starburst galaxy driving a galactic superwind. The arcsecond spatial resolution of the ACIS imaging spectroscope on the Chandra X-Ray Observatory allows us to study the spatial and spectral properties of the diffuse X-ray-emitting plasma, at a height of between 3 and 9 kpc above the disk in the northern halo of NGC 253, with greatly superior spatial and spectral resolution compared to previous X-ray instruments. We find statistically significant structure within the diffuse emission on angular scales down to ~10'' (~130 pc) and place limits on the luminosity of any X-ray-emitting "clouds" on smaller scales. 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. The spectrum of the diffuse X-ray emission is clearly thermal, although with the higher spectral resolution and sensitivity of Chandra it is clear that current simple spectral models do not provide a physically meaningful description of the spectrum. In particular, the fitted metal abundances are unphysically low. There is no convincing evidence for diffuse X-ray emission at energies above 2 keV in the halo. We show that the X-shaped soft X-ray morphology of the superwind previously revealed by ROSAT is matched by very similar X-shaped Hα 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 Hα emission. The total 0.3-2.0 keV energy band X-ray luminosity of the northern halo, LX ~ 5 × 1038 ergs s-1, is very similar to the halo Hα luminosity of LHα ~ 4 × 1038 ergs s-1, both of which are a small fraction of the estimated wind energy injection rate of ~1042 ergs s-1 from supernovae in the starburst. We show that there are a variety of models that can simultaneously explain spatially correlated X-ray and Hα emission in the halos of starburst galaxies, although the physical origin of the various emission components can be very different in different models. 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 ~ 104K gas, not only within the central kiloparsec regions of starbursts but also on ~10 kpc scales within the halos of these galaxies.
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