We discuss a new neural network-based direction of arrival estimation scheme that tackles the estimation task as a multidimensional classification problem. The proposed estimator uses a classification chain with as many stages as the number of sources. Each stage is a multiclass classification network that estimates the position of one of the sources. This approach can be interpreted as the approximation of a successive evaluation of the maximum a posteriori estimator. By means of simulations for fully sampled antenna arrays and systems with subarray sampling, we show that it is able to outperform existing estimation techniques in terms of accuracy, while maintaining a very low computational complexity.