IN THIS PAPER WE ANALYZE A VIRTUAL ELEMENT METHOD FOR THE TWO DIMENSIONAL ELASTICITY PROBLEM ALLOWING SMALL EDGES. WITH THIS APPROACH, THE CLASSIC ASSUMPTIONS ON THE GEOMETRICAL FEATURES OF THE POLYGONAL MESHES CAN BE RELAXED. IN PARTICULAR, WE CONSIDER ONLY STAR-SHAPED POLYGONS FOR THE MESHES. SUITABLE ERROR ESTIMATES ARE PRESENTED, WHERE A RIGOROUS ANALYSIS ON THE INFLUENCE OF THE LAMÉ CONSTANTS IN EACH ESTIMATE IS PRESENTED. WE REPORT NUMERICAL TESTS TO ASSESS THE PERFORMANCE OF THE METHOD.