Distributed acoustic sensing (DAS) is an emerging technology for acquiring seismic data due to its high-density and low-cost advantages. Because of the harsh acquisition environment and other unexpected reasons, the seismic signals acquired in DAS are masked by various types of complex noise, which seriously decreases the signal-to-noise ratio of seismic data. We propose a fully convolutional neural network with dense and residual connections to attenuate complex noise in DAS data. The network is designed to learn features of useful reflection signals recorded from a large number of earthquake and microseismic events, aiming at obtaining an unprecedented generalization ability. First, we generate labels using an integrated framework that attenuates specific types of noise in real DAS data, where the integrated framework includes carefully designed band-pass, structure-oriented median, and dip filters. Then, we use the patching technique to segment the training samples into many small-scale patches to reduce computational cost and improve the extraction of essential features from large-scale passive seismic data. Finally, we use the well-trained network to estimate the heavily polluted hidden signals. Compared with two advanced deep-learning methods and a traditional denoising framework, our proposed method can more effectively attenuate strong and complex noise and recover weak hidden signals in synthetic and real DAS data tests.