The combination of fingerprint positioning and 5G (the 5th Generation Mobile Communication Technology) offers broader application prospects for indoor positioning technology, but also brings challenges in real-time performance. In this paper, we propose a fingerprint positioning method based on a deep convolutional neural network (DCNN) using a classification approach in a single-base station scenario for massive multiple input multiple output-orthogonal frequency division multiplexing (MIMO-OFDM) systems. We introduce an angle-delay domain fingerprint matrix that simplifies the computation process and increases the location differentiation. The cosine distance is chosen as the fingerprint similarity criterion due to its sensitivity to angular differences. First, the DCNN model is used to determine the sub-area to which the mobile terminal belongs, and then the weighted K-nearest neighbor (WKNN) matching algorithm is used to estimate the position within the sub-area. The positioning performance is simulated in a DeepMIMO indoor environment, showing that the classification DCNN method reduces the positioning time by 77.05% compared to the non-classification method, with only a 1.08% increase in average positioning error.