Fluid antenna systems (FAS) signify a pivotal advancement in 6G communication by enhancing spectral efficiency and robustness. However, obtaining accurate channel state information (CSI) in FAS poses challenges due to its complex physical structure. Traditional methods, such as pilot-based interpolation and compressive sensing, are not only computationally intensive but also lack adaptability. Current extrapolation techniques relying on rigid parametric models do not accommodate the dynamic environment of FAS, while data-driven deep learning approaches demand extensive training and are vulnerable to noise and hardware imperfections. To address these challenges, this paper introduces a novel self-supervised learning network (SSNet) designed for efficient and adaptive channel extrapolation in FAS. We formulate the problem of channel extrapolation in FAS as an image reconstruction task. Here, a limited number of unmasked pixels (representing the known CSI of the selected ports) are used to extrapolate the masked pixels (the CSI of unselected ports). SSNet capitalizes on the intrinsic structure of FAS channels, learning generalized representations from raw CSI data, thus reducing dependency on large labelled datasets. For enhanced feature extraction and noise resilience, we propose a mix-of-expert (MoE) module. In this setup, multiple feedforward neural networks (FFNs) operate in parallel. The outputs of the MoE module are combined using a weighted sum, determined by a gating function that computes the weights of each FFN using a softmax function. Extensive simulations validate the superiority of the proposed model. Results indicate that SSNet significantly outperforms benchmark models, such as AGMAE and long short-term memory (LSTM) networks by using a much smaller labelled dataset.