In this work, an acoustic sensor network for a relative localization system is analyzed by reporting the accuracy achieved in the position estimation. The proposed system has been designed for those applications where objects are not restricted to a particular environment and thus one cannot depend on any external infrastructure to compute their positions. The objects are capable of computing spatial relations among themselves using only acoustic emissions as a ranging mechanism. The object positions are computed by a multidimensional scaling (MDS) technique and, afterwards, a least-square algorithm, based on the Levenberg-Marquardt algorithm (LMA), is applied to refine results. Regarding the position estimation, all the parameters involved in the computation of the temporary relations with the proposed ranging mechanism have been considered. The obtained results show that a fine-grained localization can be achieved considering a Gaussian distribution error in the proposed ranging mechanism. Furthermore, since acoustic sensors require a line-of-sight to properly work, the system has been tested by modeling the lost of this line-of-sight as a non-Gaussian error. A suitable position estimation has been achieved even if it is considered a bias of up to 25 of the line-of-sight measurements among a set of nodes.