With the rapid development of 5th generation (5G) communications, 5G-enabled Internet of Things (IoT) will provide high-speed and low-latency data transmissions as well as better communication coverage. 5G-enabled IoT can achieve efficient and reliable connections of massive nodes via the technical advantages of ultra-reliable low-latency communication (uRLLC), massive machine-type communication (mMTC) and enhanced mobile broadband (eMBB). However, the shortage of spectrum resources has become a bottleneck restricting the development of 5G-enabled IoT. Cognitive radio (CR) and non-orthogonal multiple access (NOMA) are emerging as promising spectrum sharing technologies to improve spectrum utilization. In this article, by integrating CR and NOMA, NOMA-based cognitive spectrum access is studied to improve the spectral efficiency of 5G-enabled IoT, by which the IoT may access the spectrum licensed to a primary user (PU) via NOMA. First, underlay, overlay and hybrid spectrum access modes based on NOMA are described, respectively, and two successive interference cancellation (SIC) schemes are proposed to guarantee the decoding performance of IoT and PU, respectively. Second, subcarrier aggregation-based NOMA and interference suppression-based NOMA decoding are presented to improve the utilization of discrete idle spectrum and decrease the impact of PU's interference on decoding, respectively. Third, grant-free NOMA and compressed sensing-based NOMA decoding are proposed to improve the performance of URLLC and mMTC, respectively. Fourth, cooperative NOMA is presented to improve the transmission performance of PU and IoT in fading channels. Finally, some open research works and challenges are discussed.