An accurate and robust localization system is crucial for autonomous vehicles (AVs) to enable safe driving in urban scenes. While existing global navigation satellite system (GNSS)-based methods are effective at locating vehicles in open-sky regions, achieving high-accuracy positioning in urban canyons such as lower layers of multi-layer bridges, streets beside tall buildings, tunnels, etc., remains a challenge. In this paper, we investigate the potential of cellular-vehicle-to-everything (C-V2X) wireless communications in improving the localization performance of AVs under GNSS-denied environments. Specifically, we propose the first roadside unit (RSU)-enabled cooperative localization framework, namely CV2X-LOCA, that only uses C-V2X channel state information to achieve lane-level positioning accuracy. CV2X-LOCA consists of four key parts: data processing module, coarse positioning module, environment parameter correcting module, and vehicle trajectory filtering module. These modules jointly handle challenges present in dynamic C-V2X networks. Extensive simulation and field experiments show that CV2X-LOCA achieves state-of-the-art performance for vehicle localization even under noisy conditions with high-speed movement and sparse RSUs coverage environments. The study results also provide insights into future investment decisions for transportation agencies regarding deploying RSUs cost-effectively.