It is well-known that the Galerkin approximation of elastic problems for nearly incompressible materials requires particular care, due to the possible occurrence of the volumetric locking phenomenon. One of the possible ways to overcome this difficulty is to adopt a mixed formulation, based on the Hellinger-Reissner functional, which implies that the elastic problem is modeled by using two a-priori independent fields: stresses and displacements. Of course, the discretization spaces for stresses and displacements must be carefully chosen, in order to avoid instabilities. In the framework of the Finite Element Method (FEM), designing stable and optimal convergent schemes for the Hellinger-Reissner functional is not at all a trivial task, even in the infinitesimal elasticity regime. A quite recent approach, the Virtual Element Method (VEM), represents an interesting alternative to FEM. Although the VEM was originally designed to deal with polytopal meshes, it was soon realized that its flexibility can be greatly beneficial in many other situations. For example, it has been successfully used in connection with the Hellinger-Reissner variational formulation. However, typical VEM needs the introduction of unphysical stabilization terms, the tuning of which might be problematic. Therefore, nowadays there is a significant effort in designing VEM schemes that avoid such terms: the so-called stabilization-free (or self-stabilized) VEM. In this talk we present a stabilization-free VEM, based on the Hellinger-Reissner functional, focusing on a simple 2D scalar problem. For quadrilateral meshes, a full stability and convergence analysis is available. We also show some numerical tests which confirm the theoretical predictions. Finally, we discuss the possibility to extend our approach to the actual elastic (2D or 3D) problem.