We present a detailed analysis of an H-2-rich, extremely strong intervening damped Ly alpha absorption system (DLA) at z(abs) = 2.786 towards the quasar J 0843+ 0221, observed with the Ultraviolet and Visual Echelle Spectrograph on the Very Large Telescope. The total column density of molecular (resp. atomic) hydrogen is log N(H-2) = 21.21 +/- 0.02 (resp. log N(H I) = 21.82 +/- 0.11), making it to be the first case in quasar absorption line studies with H2 column density as high as what is seen in (CO)-C-13-selected clouds in the Milky Way. We find that this system has one of the lowest metallicity detected among H-2-bearing DLAs, with [Zn/H] = -1.52(-0.10)(+0.08). This can be the reason for the marked differences compared to systems with similar H-2 column densities in the local Universe: (i) the kinetic temperature, T similar to 120 K, derived from the J = 0, 1 H-2 rotational levels is at least twice higher than expected; (ii) there is little dust extinction with A(V) < 0.1; (iii) no CO molecules are detected, putting a constraint on the X-CO factor X-CO > 2 x 10(23) cm(-2)/(km s(-1) K), in the very low metallicity gas. Low CO and high H-2 contents indicate that this system represents 'CO-dark/faint' gas. We investigate the physical conditions in the H-2-bearing gas using the fine-structure levels of C I, CII, Si II and the rotational levels of HD and H-2. We find the number density to be about n similar to 260-380 cm(-3), implying a high thermal pressure of 3-5 x 10(4) cm(-3) K. We further identify a trend of increasing pressure with increasing total hydrogen column density. This independently supports the suggestion that extremely strong DLAs (with log N(H)similar to 22) probe high-z galaxies at low impact parameters.