Twin-field quantum key distribution (TFQKD) systems have shown great promise for implementing practical long-distance secure quantum communication due to its measurement-device-independent nature and its ability to offer fundamentally superior rate-loss scaling than point-to-point QKD systems. A surge of research and development effort in the last two years has produced many variants of protocols and experimental demonstrations. In terms of hardware topology, TFQKD systems interfering quantum signals from two remotely phase-locked laser sources are in essence giant Mach-Zehnder interferometers (MZIs) requiring active phase stabilization. Such configurations are inherently unsuitable for a TFQKD network, where more than one user-pair share the common quantum measurement station, because it is practically extremely difficult, if not impossible, to stabilize MZIs of largely disparate path lengths, a situation that is inevitable in a multi-user-pair TFQKD network. On the other hand, Sagnac interferometer based TFQKD systems exploiting the inherent phase stability of the Sagnac ring can implement asymmetric TFQKD, and are therefore eminently suitable for implementing a TFQKD network. In this work, we experimentally demonstrate a proof-of-principle multi-user-pair Sagnac TFQKD network where three user pairs sharing the same measurement station can perform pair-wise TFQKD through time multiplexing, with channel losses up to 58 dB, and channel loss asymmetry up to 15 dB. In some cases, the secure key rates still beat the rate-loss bounds for point-to-point repeaterless QKD systems, even in this network configuration. It is to our knowledge the first multi-user-pair TFQKD network demonstration, an important step in advancing quantum communication network technologies.