Cellular user positioning is a promising service provided by Fifth Generation New Radio (5G NR) networks. Besides, Machine Learning (ML) techniques are foreseen to become an integrated part of 5G NR systems improving radio performance and reducing complexity. In this paper, we investigate ML techniques for positioning using 5G NR fingerprints consisting of uplink channel estimates from the physical layer channel. We show that it is possible to use Sounding Reference Signals (SRS) channel fingerprints to provide sufficient data to infer user position. Furthermore, we show that small fully-connected moderately Deep Neural Networks, even when applied to very sparse SRS data, can achieve successful outdoor user positioning with meter-level accuracy in a commercial 5G environment.