(abridged) We present deep archival VLT/FORS1+UVES spectroscopic observations of the system of two blue compact dwarf (BCD) galaxies SBS 0335-052E and SBS 0335-052W. Our aim is to derive element abundances in different HII regions of this unique system of galaxies and to study spatial abundance variations. We determine abundances of helium, nitrogen, oxygen, neon, sulfur, chlorine, argon and iron. The oxygen abundance in the brighter eastern galaxy varies in the range 7.11 to 7.32 in different HII regions supporting previous findings and suggesting the presence of oxygen abundance variations on spatial scales of ~1-2 kpc. The oxygen abundance in the brightest region No.1 of SBS 0335-052W is 7.22+/-0.07, consistent with previous determinations.Three other HII regions are much more metal-poor with an unprecedently low oxygen abundance of 12+logO/H=7.01+/-0.07 (region No.2), 6.98+/-0.06 (region No.3), and 6.86+/-0.14 (region No.4). These are the lowest oxygen abundances ever derived in emission-line galaxies. Helium abundances derived for the brightest HII regions of both galaxies are mutually consistent. We derive weighted mean He mass fractions of 0.2485+/-0.0012 and 0.2514+/-0.0012 for two different sets of HeI emissivities. The N/O abundance ratio in both galaxies is slightly higher than that derived for other BCDs with 12+logO/H<7.6. This implies that the N/O in extremely metal-deficient galaxies could increase with decreasing metallicity.
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