Strongly charged head-to-head (H2H) domain walls (DWs) that are purposely engineered along the [110] crystallographic orientation into ferroelectric BaTiO$_3$ single crystals have been proposed as novel 2-dimensional electron gases (2DEGs) due to their significant domain wall conductivity (DWC). Here, we quantify these 2DEG properties through dedicated Hall-transport measurements in van-der-Pauw 4-point geometry at room temperature, finding the electron mobility to reach around 400~cm$^2$(Vs)$^{-1}$, while the 2-dimensional charge density amounts to ~7$\times$10$^3$cm$^{-2}$. We underline the necessity to take account of thermal and geometrical-misalignment offset voltages by evaluating the Hall resistance under magnetic-field sweeps, since otherwise dramatic errors of several hundred percent in the derived mobility and charge density values can occur. Apart from the specific characterization of the conducting BaTiO$_3$ DW, we propose the method as an easy and fast way to quantitatively characterize ferroic conducting DWs, complementary to previously proposed scanning-probe-based Hall-potential analyses.