Quantum optical memories are a key component in a variety of quantum information applications, from extending quantum communication channels to building high-efficiency single-photon sources to enabling protocols requiring multiple synchronized qubits. However, most current photon storage systems utilize light-matter interactions and are therefore not broadband; meanwhile the available broader-bandwidth photon storage systems operate with somewhat shorter storage times or require cryogenic operation. Here we develop a system with multiplexed free-space storage cavities, able to store single photons with high efficiency over variable delays, up to 12.5 μs, and over multiple nanometers bandwidth at room temperature. The system can store multiple qubits encoded in various degrees of freedom (e.g., time-bin, and polarization) simultaneously. The work presented here has demonstrated storage of polarization states for 1.25 μs and retrieval through single-mode fiber with a state fidelity >99% and efficiency 82%.