An orthogonal basis of the Hilbert space for the quantum spin chain associated with the su(3) algebra is introduced. Such kind of basis could be treated as a nested generalization of separation of variables (SoV) basis for high-rank quantum integrable models. It is found that all the monodromy-matrix elements acting on a basis vector take simple forms. With the help of the basis, we construct eigenstates of the su(3) inhomogeneous spin torus (the trigonometric su(3) spin chain with antiperiodic boundary condition) from its spectrum obtained via the off-diagonal Bethe Ansatz (ODBA). Based on small sites (i.e. N=2) check, it is conjectured that the homogeneous limit of the eigenstates exists, which gives rise to the corresponding eigenstates of the homogenous model.
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