In a disordered fcc structure, the equiatomic Fe–Ni alloy is of great interest as a soft magnetic Invar alloy with a near zero or low thermal expansion coefficient. In the L10 ordered phase, it is of interest as a potential high-performance permanent magnet alloy. In this work, the first detailed measurements of the elastic constants in a [001]-oriented Fe-50 at. % Ni alloy single crystal under a no magnetic field condition were made using the resonant ultrasound spectroscopy technique. The elastic constants measured allow one to understand the nature of atomic bonding in a material and the mechanical response to applied stress. The single crystal sample having a rectangular parallelepiped shape was prepared from a single crystal grown using the vertical Bridgman technique. The elastic constants C11, C12, and C44 were obtained from the resonant frequencies observed over a frequency range of 100–900 kHz. Using these C11, C12, and C44 values, Young’s modulus, shear modulus, bulk modulus, and Poisson’s ratio were calculated to be 99.1 GPa, 73.4 GPa, 176.7 GPa, and 0.20, respectively. The large anisotropy factor of 3.29 highlights the strong directional dependence of the material’s mechanical behavior.