A semiclassical approach is developed for nonequilibrium quantum transport in molecular junctions. Following the early work of Miller and White [J. Chem. Phys. 84, 5059 (1986)], the many-electron Hamiltonian in second quantization is mapped onto a classical model that preserves the fermionic character of electrons. The resulting classical electronic Hamiltonian allows for real-time molecular dynamics simulations of the many-body problem from an uncorrelated initial state to the steady state. Comparisons with exact results generated for the resonant level model reveal that a semiclassical treatment of transport provides a quantitative description of the dynamics at all relevant timescales for a wide range of bias and gate potentials, and for different temperatures. The approach opens a door to treating nontrivial quantum transport problems that remain far from the reach of fully quantum methodologies.