We present deep images of a region around the Crab Nebula made with the VLA, utilizing new imaging and deconvolution algorithms in a search for a faint radio shell. The existence of a high-velocity, hydrogen-rich envelope has been predicted to account for the low total mass and kinetic energy of the observed nebula. No radio emission was detected from an extended source outside the Crab Nebula. Our limits on surface brightness are sufficiently low to rule out the existence of a shell around the Crab whose brightness is at least 2 orders of magnitude below that of SN 1006, the faintest historical shell-type supernova remnant. We consider models for the progenitor star and the presupernova environment and conclude that if a fast, outer shock exists, then it has a sharply reduced efficiency at accelerating relativistic particles from the kinetic energy of the blast wave. We also looked for a steepening of the spectral index along the boundary of the Crab Nebula itself, the signature of an outer shock. However, contrary to previous claims, no such steepening was found. The absence of any evidence at radio wavelengths that either the Crab Nebula or a hypothetical shell is interacting with the ambient medium leads to an interpretation that the supernova of A.D. 1054 was a peculiar low-energy event.