  We perform detailed spectroscopic analysis and numerical modelling of an
H2-bearing damped Lyman-alpha absorber (DLA) at zabs = 2.05 towards the quasar
FBQS J2340-0053. Metal absorption features arise from fourteen components
spread over $\Delta v_{90}$ = 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$_\sun$ ([S/H] =
-0.52+/-0.06) and dust-to-gas ratio, $\kappa$ = 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 $\sim$ 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 7,000-23,000 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.
