Directed cell motility is preceded by cell polarization - development of a front-rear asymmetry of the cytoskeleton and the cell shape. Extensive studies implicated complex spatial-temporal feedbacks between multiple signaling pathways in establishing cell polarity, yet physical mechanisms of this phenomenon remain elusive. Based on observations of lamellipodial fragments of fish keratocyte cells, we suggest a purely thermodynamic (not involving signaling) quantitative model of the cell polarization and bistability. The model is based on the interplay between pushing force exerted by F-actin polymerization on the cell edges, contractile force powered by myosin II across the cell, and elastic tension in the cell membrane. We calculate the thermodynamic work produced by these intracellular forces, and show that on the short timescale, the cell mechanics can be characterized by an effective energy profile with two minima that describe two stable states separated by an energy barrier and corresponding to the nonpolarized and polarized cells. Cell dynamics implied by this energy profile is bistable - the cell is either disk-shaped and stationary, or crescent-shaped and motile - with a possible transition between them upon a finite external stimulus able to drive the system over the macroscopic energy barrier. The model accounts for the observations of the keratocyte fragments' behavior and generates quantitative predictions about relations between the intracellular forces' magnitudes and the cell geometry and motility.