We have carried out a study of the distribution and kinematics of the neutral hydrogen in the direction of the Vela supernova remnant (SNR). A field of 6.8d x 5.4d centered at l = 264.1d, b = -1.6d was surveyed using the Parkes 64 m radio telescope (half-power beamwidth 14.7‧ at 21 cm). Nearly 2300 H i profiles were obtained with a grid spacing of 7.5‧. The presence of a thin, almost circular H i shell, centered at v = 1.6 +/- 0.8 km s^-1, is revealed. This shell delineates the outer border of the X-ray emission as shown in the ROSAT observations of Aschenbach, Egger, & Trümper and wraps around the receding part of the remnant. In addition, two higher velocity features possibly associated with Vela are observed at about -30 and 30 km s^-1. These features are interpreted as gas accelerated by the expansion of the supernova shock. The low systemic velocity observed suggests a distance shorter than 500 pc for the Vela SNR. The H i shell is ~7 deg in diameter and expands at v ~ 30 km s^-1. By assuming a distance of 350 pc, we calculate for this shell a linear radius of 22 pc, a swept-up mass of ~1200-2300 M_⊙, and an atomic preshock density of ~1-2 cm^-3. The kinetic energy transferred by the supernova shock into the interstellar medium is ~(1-2) x 10^49 ergs, while the initial energy of the explosion is estimated to be ~(1-2.5) x 10^51 ergs. We present the distribution of the column density of the neutral material absorbing the X-radiation, an essential parameter in the analysis of X-ray data. A comparison between the H i and Hα emission suggests that the H i shell contains embedded dust that might be responsible for increased optical absorption in this region. On the other hand, the brightest arc-shaped optical filaments associated with the western side of Vela show good correspondence with the H i features. From a comparison between the H i and Molonglo Observatory Synthesis Telescope 843 MHz radio continuum emission, we find that the outermost arched radio filaments correlate well with the main ridge of the H i shell. No strong inhomogeneities were found in the ambient H i medium in the direction of Vela X (the central nebula, powered by the pulsar PSR B0833-45).