The meshless method based on the local boundary integral equation (LBIE) is a promising method for solving boundary value problems, using an local unsymmetric weak form and shape functions from the moving least squares approximation. In the present paper, the meshless method based on the LBIE for solving problems in linear elasticity is developed and numerically implemented. The present method is a truly meshless method, as it does not need a `finite element mesh', either for purposes of interpolation of the solution variables, or for the integration of the energy. All integrals in the formulation can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. The essential boundary conditions in the present formulation can be easily imposed even when the non-interpolative moving least squares approximation is used. Several numerical examples are presented to illustrate the implementation and performance of the present method. The numerical examples show that high rates of convergence with mesh refinement for the displacement and energy norms are achievable. No post-processing procedure is required to compute the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.