The theoretically proven security of quantum key distribution (QKD) could revolutionise how information exchange is protected in the future. Several field tests of QKD have proven it to be a reliable technology for cryptographic key exchange and have demonstrated nodal networks of point-to-point links. However, so far no convincing answer has been given to the question of how to extend the scope of QKD beyond niche applications in dedicated high security networks. Here we show that adopting simple and cost-effective telecommunication technologies to form a quantum access network can greatly expand the number of users in quantum networks and therefore vastly broaden their appeal. We are able to demonstrate that a high-speed single-photon detector positioned at a network node can be shared between up to 64 users for exchanging secret keys with the node, thereby significantly reducing the hardware requirements for each user added to the network. This point-to-multipoint architecture removes one of the main obstacles restricting the widespread application of QKD. It presents a viable method for realising multi-user QKD networks with resource efficiency and brings QKD closer to becoming the first widespread technology based on quantum physics.