Electronic transport properties in the spin-orbit Mott insulator Ba2IrO4 were studied under high pressure (P), up to 15 GPa. Resistivity measurements revealed a metal-insulator (M-I) transition at 13.8 GPa in Ba2IrO4. In the insulating state (P < 13.8 GPa), the resistivity well follows the variable-range-hopping regime, suggesting that the carrier conduction is dominated by Anderson localization. In the metallic state (P >= 13.8 GPa), non-Fermi-liquid behavior was observed at low temperature. Ba2IrO4 exhibits an unconventional critical exponent (delta = 1-2) at the M-I transition. These results suggest that Ba2IrO4 has an unusual electronic state affected by the "marginal quantum critical point (MQCP)."