The realization of a network of quantum registers is an outstanding challenge in quantum science and technology. We experimentally investigate a network node that consists of a single nitrogen-vacancy center electronic spin hyperfine coupled to nearby nuclear spins. We demonstrate individual control and readout of five nuclear spin qubits within one node. We then characterize the storage of quantum superpositions in individual nuclear spins under repeated application of a probabilistic optical internode entangling protocol. We find that the storage fidelity is limited by dephasing during the electronic spin reset after failed attempts. By encoding quantum states into a decoherence-protected subspace of two nuclear spins, we show that quantum coherence can be maintained for over 1000 repetitions of the remote entangling protocol. These results and insights pave the way towards remote entanglement purification and the realization of a quantum repeater using nitrogen-vacancy center quantum-network nodes.