This paper provides an initial investigation on the application of convolutional neural networks (CNNs) for fingerprint-based positioning using measured massive MIMO channels. When represented in appropriate domains, massive MIMO channels have a sparse structure which can be efficiently learned by CNNs for positioning purposes. We evaluate the positioning accuracy of state-of-the-art CNNs with channel fingerprints generated from a channel model with a rich clustered structure: the COST 2100 channel model. We find that moderately deep CNNs can achieve fractional-wavelength positioning accuracies, provided that an enough representative data set is available for training.