We report the synthesis, crystal structure, and magnetic properties of the two new quantum antiferromagnets A3ReO5Cl2 (A = Sr, Ba). The crystal structure is isostructural with the mineral pinalite Pb3WO5Cl2, in which the Re6+ ion is square pyramidally coordinated by five oxide atoms and forms an anisotropic triangular lattice (ATL) made of S = 1/2 spins. The magnetic interactions J and J' in the ATL are estimated from magnetic susceptibilities to be 19.5 (44.9) and 9.2 (19.3) K, respectively, with J'/J = 0.47 (0.43) for A = Ba (Sr). For each compound, the heat capacity at low temperatures shows a large T-linear component with no signature of long-range magnetic order above 2 K, which suggests a gapless spin liquid state of one-dimensional character of the J chains in spite of the significantly large J' couplings. This is a consequence of one-dimensionalization by geometrical frustration in the ATL magnet; a similar phenomenon has been observed in two compounds with slightly smaller J'/J values: Cs2CuCl4 (J'/J = 0.3) and the related compound Ca3ReO5Cl2 (0.32). Our findings demonstrate that 5d mixed-anion compounds provide a unique opportunity to explore novel quantum magnetism.