Quantum teleportation, the faithful transfer of an unknown input state onto a remote quantum system1, is a key component in long-distance quantum communication protocols2 and distributed quantum computing3,4. At the same time, high-frequency nano-optomechanical systems5 hold great promise as nodes in a future quantum network6, operating on-chip at low-loss optical telecom wavelengths with long mechanical lifetimes. Recent demonstrations include entanglement between two resonators7, a quantum memory8 and microwave-to-optics transduction9,10,11. Despite these successes, quantum teleportation of an optical input state onto a long-lived optomechanical memory is an outstanding challenge. Here we demonstrate quantum teleportation of a polarization-encoded optical input state onto the joint state of a pair of nanomechanical resonators. Our protocol also allows to store and retrieve an arbitrary qubit state onto a dual-rail encoded optomechanical quantum memory. This work demonstrates the full functionality of a single quantum repeater node and presents a key milestone towards applications of optomechanical systems as quantum network nodes.