We report on high-resolution x-ray synchrotron powder-diffraction, magnetic-susceptibility, sound-velocity, thermal-expansion, and heat-capacity studies of the stoichiometric spinel FeCr₂S₄. We provide clear experimental evidence of a structural anomaly which accompanies an orbital-order transition at low temperatures due to a static cooperative Jahn-Teller effect. At 9 K, magnetic susceptibility, ultrasound velocity, and specific heat reveal pronounced anomalies that correlate with a volume contraction as evidenced by thermal-expansion data. The analysis of the low-temperature heat capacity using a mean-field model with a temperature-dependent gap yields a gap value of about 18 K and is interpreted as the splitting of the electronic ground state of Fe²⁺ by a cooperative Jahn-Teller effect. This value is close to the splitting of the ground state due to spin-orbit coupling for isolated Fe²⁺ ions in an insulating matrix, indicating that Jahn-Teller and spin-orbit coupling are competing energy scales in this system. We argue that due to this competition, the spin-reorientation transition at around 60 K marks the onset of short-range orbital ordering accompanied by a clear broadening of Bragg reflections, an enhanced volume contraction compared to usual anharmonic behavior, and a softening of the lattice observed in the ultrasound measurements.