We present a detailed analysis of an H 2 -rich, extremely strong intervening damped Ly-α 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 H 2 column density as high as what is seen in 13 CO-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 +0.08 −0.10 . 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 ∼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 × 10 23 cm −2 /(km/s 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, C ii, Si ii and the rotational levels of HD and H 2 . We find the number density to be about n ∼ 260 − 380 cm −3 , implying a high thermal pressure of 3 − 5 × 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) ∼ 22) probe high-z galaxies at low impact parameters.