Ultra-narrowband (UNB) communications has become a signature feature for many emerging low-power wide-area (LPWA) networks. Specifically, using extremely narrowband signals helps the network connect more Internet-of-things (IoT) devices within a given band. It also improves robustness to interference, extending the coverage of the network. In this paper, we study the coexistence capability of UNB networks and their scalability to enable massive access. To this end, we develop a stochastic geometry framework to analyze and model UNB networks on a large scale. The framework captures the unique characteristics of UNB communications, including the asynchronous time-frequency access, signal repetition, and the absence of base station (BS) association. Closed-form expressions of the transmission success probability and network connection density are presented for several UNB protocols. We further discuss multiband access for UNB networks, proposing a low-complexity protocol. Our analysis reveals several insights on the geographical diversity achieved when devices do not connect to a single BS, the optimal number of signal repetitions, and how to utilize multiple bands without increasing the complexity of BSs. Simulation results are provided to validate the analysis, and they show that UNB communications enables a single BS to connect thousands of devices even when the spectrum is shared with other networks.