We present observations of the dark gamma-ray burst GRB 051008 provided by Swift/BAT, Swift/XRT, Konus-WIND, INTEGRAL/SPI-ACS in the high-energy domain and the Shajn, Swift/UVOT, Tautenburg, NOT, Gemini and Keck I telescopes in the optical and near-infrared bands. The burst was detected only in gamma- and X-rays and neither a prompt optical nor a radio afterglow was detected down to deep limits. We identified the host galaxy of the burst, which is a typical Lyman-break galaxy (LBG) with R-magnitude of 24.06 ± 0.10 mag. A redshift of the galaxy of z = 2.77_{-0.20}^{+0.15} is measured photometrically due to the presence of a clear, strong Lyman-break feature. The host galaxy is a small starburst galaxy with moderate intrinsic extinction (AV = 0.3) and has a star formation rate of ˜60 M⊙ yr-1 typical for LBGs. It is one of the few cases where a GRB host has been found to be a classical LBG. Using the redshift we estimate the isotropic-equivalent radiated energy of the burst to be Eiso = (1.15 ± 0.20) × 1054 erg. We also provide evidence in favour of the hypothesis that the darkness of GRB 051008 is due to local absorption resulting from a dense circumburst medium.