A face-centred finite volume (FCFV) method is proposed for linear elastostatic problems. The FCFV is a mixed hybrid formulation, featuring a system of first-order equations, that defines the unknowns on the faces (edges in two dimensions) of the mesh cells. The symmetry of the stress tensor is strongly enforced using the well-known Voigt notation and the displacement and stress fields inside each cell are obtained by means of explicit formulas. The resulting FCFV method is robust and locking-free in the nearly incompressible limit. Numerical experiments in two and three dimensions show optimal convergence of the displacement and the stress fields without any reconstruction. Moreover, the accuracy of the FCFV method is not sensitive to mesh distortion and stretching. Classical benchmark tests including Kirch’s plate and Cook’s membrane problems in two dimensions as well as three dimensional problems involving shear phenomenons, pressurised thin shells and complex geometries are presented to show the capability and potential of the proposed methodology.