Quantum memories capable of faithfully storing and recalling quantum states on-demand are powerful ingredients in bulding quantum networks [arXiv:0806.4195] and quantum information processors [arXiv:1109.3743]. As in conventional computing, key attributes of such memories are high storage density and, crucially, random access, or the ability to read from or write to an arbitrarily chosen register. However, achieving such random access with quantum memories [arXiv:1904.09643] in a dense, hardware-efficient manner remains a challenge, for example requiring dedicated cavities per qubit [arXiv:1109.3743] or pulsed field gradients [arXiv:0908.0101]. Here we introduce a protocol using chirped pulses to encode qubits within an ensemble of quantum two-level systems, offering both random access and naturally supporting dynamical decoupling to enhance the memory lifetime. We demonstrate the protocol in the microwave regime using donor spins in silicon coupled to a superconducting cavity, storing up to four multi-photon microwave pulses and retrieving them on-demand up to 2~ms later. A further advantage is the natural suppression of superradiant echo emission, which we show is critical when approaching unit cooperativity. This approach offers the potential for microwave random access quantum memories with lifetimes exceeding seconds [arXiv:1301.6567, arXiv:2005.09275], while the chirped pulse phase encoding could also be applied in the optical regime to enhance quantum repeaters and networks.