Two Chandra observations have been used to search for thermal X-ray emission from within and around the Crab Nebula. Dead time was minimized by excluding the brightest part of the nebula from the field of view. A dust-scattered halo comprising 5% of the strength of the Crab is clearly detected, with surface brightness measured out to a radial distance of 18'. Coverage is 100% at 4', 50% at 12', and 25% at 18'. The observed halo is compared with predictions based on three different interstellar grain models, and one can be adjusted to fit the observation. This dust halo and mirror scattering form a high background region that has been searched for emission from shock-heated material in an outer shell. We find no evidence for such emission. We can set upper limits a factor of 10-1000 less than the surface brightness observed from outer shells around similar remnants. The upper limit for X-ray luminosity of an outer shell is ~1034 ergs s-1. Although it is possible to reconcile our observation with an 8-13 Msolar progenitor, we argue that this is unlikely.