In frequency-division duplexing (FDD) massive multiple-input multiple-output (MIMO) systems, downlink channel state information (CSI) needs to be sent back to the base station (BS) by the users, which causes prohibitive feedback overhead. In this paper, we propose a lightweight and flexible deep learning-based CSI feedback approach by capitalizing on deep equilibrium models. Different from existing deep learning-based methods that stack multiple explicit layers, we propose an implicit equilibrium block to mimic the behavior of an infinite-depth neural network. In particular, the implicit equilibrium block is defined by a fixed-point iteration and the trainable parameters in different iterations are shared, which results in a lightweight model. Furthermore, the number of forward iterations can be adjusted according to users' computation capability, enabling a flexible accuracy-efficiency trade-off. Simulation results will show that the proposed design obtains a comparable performance as the benchmarks but with much-reduced complexity and permits an accuracy-efficiency trade-off at runtime.