Full-duplex (FD) has emerged as a disruptive communications paradigm for enhancing the achievable spectral efficiency (SE), thanks to the recent major breakthroughs in self-interference (SI) mitigation. The FD versus half-duplex (HD) SE gain, in cellular networks, is however largely limited by the mutual-interference (MI) between the downlink (DL) and the uplink (UL). A potential remedy for tackling the MI bottleneck is through cooperative communications. This paper provides a stochastic design and analysis of FD enabled cloud radio access network (C-RAN) under the Poisson point process (PPP)-based abstraction model of multi-antenna radio units (RUs) and user equipments (UEs). We consider different disjoint and user-centric approaches towards the formation of finite clusters in the C-RAN. Contrary to most existing studies, we explicitly take into consideration non-isotropic fading channel conditions and finite-capacity fronthaul links. Accordingly, upper-bound expressions for the C-RAN DL and UL SEs, involving the statistics of all intended and interfering signals, are derived. The performance of the FD C-RAN is investigated through the proposed theoretical framework and Monte-Carlo (MC) simulations. The results indicate that significant FD versus HD C-RAN SE gains can be achieved, particularly in the presence of sufficient-capacity fronthaul links and advanced interference cancellation capabilities.