  The discovery of the anomalous Hall effect (AHE) in bulk metallic
antiferromagnets (AFMs) motivates the search of the same phenomenon in
two-dimensional (2D) systems, where a quantized anomalous Hall conductance can
in principle be observed. Here, we present experiments on micro-fabricated
devices based on Co$_{1/3}$NbS$_2$, a layered AFM that was recently found to
exhibit AHE in bulk crystals below the N\'eel temperature T$_N$ = 29 K.
Transport measurements reveal a pronounced resistivity anisotropy, indicating
that upon lowering temperature the electronic coupling between individual
atomic layers is increasingly suppressed. The experiments also show an
extremely large anomalous Hall conductivity of approximately 400 S/cm, more
than one order of magnitude larger than in the bulk, which demonstrates the
importance of studying the AHE in small exfoliated crystals, less affected by
crystalline defects. Interestingly, the corresponding anomalous Hall
conductance, when normalized to the number of contributing atomic planes, is
$\sim \, 0.6 \; e^2/h$ per layer, approaching the value expected for the
quantized anomalous Hall effect. The observed strong anisotropy of transport
and the very large anomalous Hall conductance per layer make the properties of
Co$_{1/3}$NbS$_2$ compatible with the presence of partially filled
topologically non-trivial 2D bands originating from the magnetic superstructure
of the antiferromagnetic state. Isolating atomically thin layers of this
material and controlling their charge density may therefore provide a viable
route to reveal the occurrence of the quantized AHE in a 2D AFM.
