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We perform detailed spectroscopic analysis and numerical modelling of an H₂-bearing damped Lyman-α absorber (DLA) at z_abs = 2.05 towards the quasar FBQS J2340-0053. Metal absorption features arise from 14 components spread over Δv₉₀ = 114 km s^–1, seven of which harbour H₂. Column densities of atomic and molecular species are derived through Voigt profile analysis of their absorption lines. We measure total N(H I), N(H₂), and N(HD) to be 20.35 ± 0.05, 17.99 ± 0.05, and 14.28 ± 0.08 (log cm^–2), respectively. H₂ is detected in the lowest six rotational levels of the ground vibrational state. The DLA has metallicity, Z = 0.3 Z_☉ ([S/H] = -0.52 ± 0.06) and dust-to-gas ratio, κ = 0.34 ± 0.07. Numerical models of the H₂ components are constrained individually to understand the physical structure of the DLA. We conclude that the DLA is subjected to the metagalactic background radiation and cosmic ray ionization rate of ∼10^–15.37 s^–1. Dust grains in this DLA are smaller than grains in the Galactic interstellar medium. The inner molecular regions of the H₂ components have density, temperature, and gas pressure in the range 30-120 cm^–3, 140-360 K, and 7000-23 000 cm^–3 K, respectively. Micro-turbulent pressure is a significant constituent of the total pressure, and can play an important role in these innermost regions. Our H₂ component models enable us to constrain component-wise N(H I), and elemental abundances of sulphur, silicon, iron, and carbon. We deduce the line-of-sight thickness of the H₂-bearing parts of the DLA to be 7.2 pc.