We employ a self-similar Laplacian in the one-dimensional infinite space and deduce a model for one-dimensional self-similar elasticity. As a consequence of self-similarity this Laplacian assumes the non-local form of a self-adjoint combination of fractional integrals. The linear elastic constitutive law becomes a non-local convolution with the elastic modulus function being a power-law kernel. We outline some principal features of a linear self-similar elasticity theory in one dimension. We find an anomalous behavior of the elastic modulus function reflecting a regime of critically slowly decreasing interparticle interactions in one dimension. The approach can be generalized to the n dimensional physical space (Michelitsch, Maugin, Nowakowski, Nicolleau, & Rahman, to be published).