This paper addresses the output tracking control for a class of non-affine nonlinear systems in pure-feedback form with unknown dynamics subject to full state constraints and unknown external time-varying bounded disturbances. This problem is challenging due to the non-affine structure coupling with the constraints on states and external disturbances. To overcome the problem, the mean value theorem is applied to deduce the variables with affine appearance in the system dynamics, and the integral Barrier Lyapunov Functionals (iBLFs) are utilized in the recursive backstepping design to handle the unknown deduced control gains and state constraints together, and adapting parameters are developed to estimate the unknown bounds on neural networks (NNs) reconstruction errors and external disturbances. Through Lyapunov synthesis, all signals in the closed-loop system are proved to be semi-globally uniformly ultimately bounded (SGUUB), and all state constraints are guaranteed provided that the states are initially in the constrained regions, and tracking errors are ensured to converge to an adjustable neighborhood of desired trajectory. An application to Van der Pol oscillator with state constraints illustrates the performance of proposed control.