Molecular simulations are used to elucidate compressive failure in polyethylene fibers. Compression of perfect polyethylene crystals is found to give rise to a long wavelength Euler buckling instability. The critical stress necessary to produce this buckling instability decreases as the wavelength of the instability increases, and it approaches the value of the lowest shear modulus in the limit of very long wavelength. The role of defects and the lamellar structure on the compressive failure mechanism of real polyethylene fibers is qualitatively addressed by simulations of n-alkane crystals. In contrast to the infinite chain systems, elastic instabilities in n-alkane crystals occur at stresses significantly below the lowest sheer modulus; this reduction in the critical stress occurs because the instability allows the relaxation of intrachain distortions concentrated near chain ends which accumulate during compression. Similar elastic instabilities associated with defects and the lamellar structure in real polyethylene fibers could explain the experimental observation that the compressive strength of polyethylene is significantly lower than the measured shear modulus.