  By means of Shubnikov-de-Haas and de-Haas-van-Alphen oscillations, and ab
initio calculations, we have studied the Fermi surface of high-quality
BaNiS$_2$ single crystals, with mean free path $l \sim 400 ~\text{\AA}$. The
angle and temperature dependence of quantum oscillations indicates a
quasi-two-dimensional Fermi surface, made of an electron-like tube centred at
$\Gamma$, and of 4 hole-like cones, generated by Dirac bands, weakly dispersive
in the out-of-plane direction. Ab initio electronic structure calculations, in
the density functional theory framework, show that the inclusion of screened
exchange is necessary to account for the experimental Fermi pockets. Therefore,
the choice of the functional becomes crucial. A modified HSE hybrid functional
with 7% of exact exchange outperforms both GGA and GGA+U density functionals,
signalling the importance of non-local screened-exchange interactions in
BaNiS$_2$, and, more generally, in $3d$ compensated semimetals.
