  We present a study of ~100 high redshift (z~2-4) extremely strong damped
Lyman-alpha systems (ESDLA, with N(HI)>0.5x10^22 cm^-2) detected in quasar
spectra from the Baryon Oscillation Spectroscopic Survey SDSS-III DR11. We
study the neutral hydrogen, metal, and dust content of this elusive population
of absorbers and confirm our previous finding that the high column density end
of the N(HI) frequency distribution has a relatively shallow slope with
power-law index -3.6, similar to what is seen from 21-cm maps in nearby
galaxies. The stacked absorption spectrum indicates a typical metallicity
~1/20th solar, similar to the mean metallicity of the overall DLA population.
The relatively small velocity extent of the low-ionisation lines suggests that
ESDLAs do not arise from large-scale flows of neutral gas. The high column
densities involved are in turn more similar to what is seen in DLAs associated
with gamma-ray burst afterglows (GRB-DLAs), which are known to occur close to
star forming regions. This indicates that ESDLAs arise from lines of sight
passing at very small impact parameters from the host galaxy, as observed in
nearby galaxies. This is also supported by simple theoretical considerations
and recent high-z hydrodynamical simulations. We strongly substantiate this
picture by the first statistical detection of Lya emission with
<L>~(0.6+/-0.2)x10^42 erg/s in the core of ESDLAs (corresponding to about 0.1
L* at z~2-3), obtained through stacking the fibre spectra (of radius 1"
corresponding to ~8 kpc at z~2.5). [truncated]
