We report on the microscopic magnetic model of a spin-1/2 magnet BaV_3O_8. In contrast to earlier phenomenological analysis, our density-functional band-structure calculations combined with quantum Monte-Carlo simulations establish a relatively simple and non-frustrated model of weakly coupled spin chains with the intrachain coupling of J∼ 38 K and the Neel temperature of T_N∼ 6 K, both in excellent agreement with the experiment. The intrachain coupling between the spin-1/2 V^+4 ions takes

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... via two contiguous V^+5O_4 tetrahedra forming an extended superexchange pathway with the V^+4--V^+4 distance of 7.44 A. Surprisingly, this pathway is preferred over shorter V^+4--V^+4 connections, owing to peculiarities in the interacting orbitals of the magnetic V^+4 ions and V^+5 ions that are non-magnetic, but feature low-lying 3d states contributing to the superexchange process. We also note that the crystal structure of BaV_3O_8 supports the long-sought uniform arrangement of Dzyaloshinsky-Moriya (DM) couplings on a spin-1/2 chain. While our calculations yield only a weak DM anisotropy in BaV_3O_8, the crystal structure of this compound provides a suitable framework for the search of spin chains with the uniform DM anisotropy in other compounds of the vanadate family.