VLA and Parkes 64 m radiotelescope 21-cm observations of the starburst dwarf galaxy NGC 5253 reveal a multi-component non-axisymmetric HI distribution. The component associated with the stellar body shows evidence for a small amount of rotational support aligned with the major axis, in agreement with optically measured kinematics and consistent with the small galaxian mass. Approximately 20-30% of the HI emission is associated with a second component, an HI plume extending along the optical minor axis to the southeast. We consider outflow, inflow, and tidal origins for this feature. Outflow appears improbable, inflow is a possibility, and tidal debris is most consistent with the observations. These observations also reveal a filamentary third component that includes an 800 pc diameter HI shell or bubble to the west of the nucleus, coinciding with an Halpha shell. The mass of HI in the shell may be as large as ~4x10^6 Msun. This large mass, coupled with the lack of expansion signatures in the neutral and ionized gas (v<30 km/s), suggests that this feature may be an example of a starburst-blown bubble stalled by interaction with a massive neutral envelope. Many other HI kinematic features closely resemble those seen in Halpha emission from the ionized gas, supporting the interpretation of neutral and ionized gas outflow at velocities of ~30 km/s. Comparison between extinction estimates from the Balmer emission-line decrement and the HI column densities suggest a gas-to-dust ratio 2-3 times the Galactic value in this low-metallicity (Z=1/4 Zsun) galaxy.