The nine independent single-crystal elastic constants of a new ceramic, titanium tetraboride (Ti3B4), have been determined by first principles calculations, and the data were validated experimentally through nanoindentation testing. The independent elastic constants, which are specific to the crystal structure, are important for the fundamental characterization of mechanical and physical properties of the group of hard compounds such as the transition metal borides. The elastic constants of Ti3B4 were determined from crystal strain energies that were calculated by applying specific deformations within WIEN2k platform utilizing full-potential linear augmented plane wave (FLAPW) and generalized gradient approximation (GGA). The WIEN2K package is based on all-electron calculations, and hence is considered as the most accurate for first principles calculations. It has been found that the polycrystalline elastic moduli, determined as the Voigt-Reuss-Hill averages of the independent elastic constants, are quite impressive (E = 492 GPa, G = 217 GPa, B = 224 GPa, = 0.13) placing the tetraboride very close to the well-known titanium diboride (E = 570 GPa, G = 254 GPa, B = 249 GPa, = 0.12). The strong B-B chains were found to be largely responsible for the high values of elastic stiffness constants, in particular the c(33) describing stiffness in the [001] direction. The electron charge densities were found to be accumulated to a higher degree along the B-B bonds, resulting in strengthening of the B-B chains in the lattice. The promising data motivated the first experimental synthesis of Ti3B4 in a bulk form, which is also described in this work. To validate the elastic constants determined from first principles, elastic moduli were determined by nanoindentations in multiple grains of a polycrystalline Ti3B4, synthesized by electric field-activated reaction sintering. The range of elastic moduli determined from nanoindentation was found to agree well with the range determined by computation. The calculations and experiments demonstrate that Ti3B4 has the potential to be one of the leading structural ceramics.