We present the results of HI mapping with the NRAO VLA of one of the most metal-deficient blue compact dwarf (BCD) galaxies known, SBS 0335-052, with an oxygen abundance of only 1/40 that of the Sun. We study the structure and dynamics of the neutral gas in this chemically young object with a spatial resolution of 20.5"x15" (~ 5.4 x 3.9 kpc at an assumed distance of 54.3 Mpc), a sensitivity at the 2sigma detection level of ~2.0K or 7.5x10^19 cm^-2 and a velocity resolution of 21.2 km s^-1. We detected a large HI complex associated with this object with an overall size of about 66 by 22 kpc and elongated in the East-West direction. There are two prominent, slightly resolved peaks visible in the integrated HI map, separated in the East-West direction by 22 kpc (84"). The eastern peak is nearly coincident with the position of the optical galaxy SBS 0335-052. The western peak is about a factor of 1.3 brighter in the HI line and is identified with a faint blue compact dwarf galaxy, SBS 0335-052W, with m_B = 19.4, and a metallicity close to the lowest values known for BCDs, about 1/50 that of the Sun. The radial velocities of both systems are similar, suggesting that the two BCDs SBS 0335-052 and SBS 0335-052W constitute a pair of dwarf galaxies embedded in a common HI envelope. Alternatively, the BCDs can be the nuclei of two distinct interacting primordial HI clouds. The estimated total dynamical mass, assuming the BCDs form a bound system, is larger than 6x10^9 Msun. This is to be compared to a total gaseous mass M_gas = 2.1x10^9 Msun, and a total stellar mass Mstar < 10^8 Msun. Hence, the mass of the SBS 0335-052 system is dominated by dark matter (abridged).