In this paper, we propose a novel deep unsupervised learning-based approach that jointly optimizes antenna selection and hybrid beamforming to improve the hardware and spectral efficiencies of massive multiple-input-multiple-output (MIMO) downlink systems. By employing ResNet to extract features from the channel matrices, two neural networks, i.e., the antenna selection network (ASNet) and the hybrid beamforming network (BFNet), are respectively proposed for dynamic antenna selection and hybrid beamformer design. Furthermore, a deep probabilistic subsampling trick and a specially designed quantization function are respectively developed for ASNet and BFNet to preserve the differentiability while embedding discrete constraints into the network structures. With the aid of a flexibly designed loss function, ASNet and BFNet are jointly trained in a phased unsupervised way, which avoids the prohibitive computational cost of acquiring training labels in supervised learning. Simulation results demonstrate the advantage of the proposed approach over conventional optimization-based algorithms in terms of both the achieved rate and the computational complexity.