Long-slit spectra of two peripheral regions around the Crab Nebula show no Hα emission down to a flux level of 1.5 x 10(-7) erg cm(-2) s(-1) sr(-1) (0.63 Rayleigh), corresponding to an emission measure limit of 4.2 cm(-6) pc (3sigma ) assuming A_V = 1.6(m) and T_e = 7000 K. This is below the flux levels reported by Murdin & Clark [Nature, 294, 543 (1981)] for a Hα halo around the Crab. Narrow Hβ emission as described by Murdin [MNRAS, 269, 89 (1994)] is detected but appears to be galactic emission unassociated with the remnant. A review of prior searches indicates no convincing observational evidence to support either a high- or low-velocity envelope around the remnant. Spectral scans confirm a well-organized, N-S expansion asymmetry of the filaments with a ~ 500 km s(-1) central velocity constriction as described by MacAlpine et al. [ApJ, 342, 364 (1989)] and Lawrence et al. [AJ, 109, 2635 (1995)] but questioned by Hester et al. [ApJ, 448, 240 (1995)]. The velocity pinching appears to coincide with an east-west chain of bright [O III] and helium-rich filaments. This expansion asymmetry might be the result of ejecta interaction with a disk of circumstellar matter, but such a model may be inconsistent with H and He filament abundances in the velocity constriction zone. A re-analysis of the remnant's total mass suggests that the filaments contain 4.6 +/- 1.8 Msun in ionized and neutral gas, about twice that of earlier estimates. For a 10 Msun progenitor, this suggests that =~ 4 Msun remains to be detected in an extended halo or wind.