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While federated learning has shown strong results in optimizing a machine learning model without direct access to the original data, its performance may be hindered by intermittent client availability which slows down the convergence and biases the final learned model. There are significant challenges to achieve both stable and bias-free training under arbitrary client availability. To address these challenges, we propose a framework named Federated Graph-based Sampling (FedGS), to stabilize the global model update and mitigate the long-term bias given arbitrary client availability simultaneously. First, we model the data correlations of clients with a Data-Distribution-Dependency Graph (3DG) that helps keep the sampled clients data apart from each other, which is theoretically shown to improve the approximation to the optimal model update. Second, constrained by the far-distance in data distribution of the sampled clients, we further minimize the variance of the numbers of times that the clients are sampled, to mitigate long-term bias. To validate the effectiveness of FedGS, we conduct experiments on three datasets under a comprehensive set of seven client availability modes. Our experimental results confirm FedGS's advantage in both enabling a fair client-sampling scheme and improving the model performance under arbitrary client availability. Our code is available at \url{https://github.com/WwZzz/FedGS}.