Single crystals of the quantum low-dimensional antiferromagnet Rb4Cu(MoO4)(3) and the nonmagnetic analogue Ru4Zn(MoO4)(3) have been synthesized by a flux-growth method. Detailed structural studies indicate that the Cu(II)-O network separated by a Moat layer has a strongly anisotropic hybridization along the a-axis, forming a quasi-one-dimensional (1-d) chain of Cu(II) S = 1/2 spins. Furthermore, our low-temperature thermodynamic measurements have revealed that a quantum paramagnetic state with Wilson ratio similar to 2 remains stable down to at least 0.1 K, 100 times lower than the intrachain antiferromagnetic coupling scale. The low-temperature magnetic and thermal properties are found to be consistent with theoretical predictions made for a 1-d network of S = 1/2 spins.