The mechanical properties of oxalic acid dihydrate and anhydrous oxalic acid (α and β polymorphic forms) were obtained by using rigorous theoretical solid-state methods based on density functional theory using plane waves and pseudopotentials. The calculated crystal structures and X-ray powder diffraction patterns of these materials were found to be in excellent agreement with the experimental information. Since the calculated elasticity matrices fullfilled the Born stability conditions, the corresponding crystal structures were found to be mechanically stable. A large number of relevant mechanical properties including the values of the bulk moduli and their pressure derivatives, shear and Young moduli, Poisson ratios, ductility and hardness indices, and mechanical anisotropy values of these materials were reported. The three forms of oxalic acid are highly anisotropic ductile materials having low hardness and bulk moduli. The three materials are shown to display small negative Poisson ratios (NPR) and to exhibit the phenomenon of negative linear compressibility (NLC) for applied pressures along the direction of the minimum Poisson ratio. In addition, they undergo pressure induced phase transitions for relatively small applied pressures. The analysis of the crystal structures of these materials as a function of pressure demonstrates that the mechanism of NLC of these materials is unrelated to the wine-rack structural mechanism commonly used to rationalize this phenomenon. The three forms of oxalic acid considered in this work are molecular crystals whose structures are characterized by structural elements which are not directly bonded but held together by weak van der Waals forces. The weak bonding between these elements is able to accommodate the structural variations originating from the application of pressure, but the resulting structural deformations appear to be counterintuitive and lead to the anomalous mechanical behavior of these materials.