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 microfabricated devices based on Co1/3NbS2, a layered AFM that was recently found to exhibit AHE in bulk crystals below the Neel 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 similar to 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 Co1/3NbS2 compatible with the presence of partially filled topologically nontrivial 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.