In this paper we shall establish necessary and sufficient conditions for a system of characteristic vectors of a nonunitary operator in Hilbert space to form an unconditional basis (i.e., a basis similar to an orthonormal basis). These conditions will be formulated in terms of the characteristic function of the operator and its functional model, as developed by B. S. Nagy and C. Foias [1]. Therefore, we shall consider only operators of contraction, which are in some (sufficiently weak) sense close to unitary, i.e., operators which are well-placed in the scheme of Nagy and Foias.