The XENONnT detector uses the latest and largest liquid xenon-based time projection chamber (TPC) operated by the XENON Collaboration, aimed at detecting Weakly Interacting Massive Particles and conducting other rare event searches. The XENONnT data acquisition (DAQ) system constitutes an upgraded and expanded version of the XENON1T DAQ system. For its operation, it relies predominantly on commercially available hardware accompanied by open-source and custom-developed software. The three constituent subsystems of the XENONnT detector, the TPC (main detector), muon veto, and the newly introduced neutron veto, are integrated into a single DAQ, and can be operated both independently and as a unified system. In total, the DAQ digitizes the signals of 698 photomultiplier tubes (PMTs), of which 253 from the top PMT array of the TPC are digitized twice, at ×10 and ×0.5 gain. The DAQ for the most part is a triggerless system, reading out and storing every signal that exceeds the digitization thresholds. Custom-developed software is used to process the acquired data, making it available within ∼30 s for live data quality monitoring and online analyses. The entire system with all the three subsystems was successfully commissioned and has been operating continuously, comfortably withstanding readout rates that exceed ∼500 MB/s during calibration. Livetime during normal operation exceeds 99% and is ∼90% during most high-rate calibrations. The combined DAQ system has collected more than 2 PB of both calibration and science data during the commissioning of XENONnT and the first science run.