The Deep Underground Neutrino Experiment (DUNE) could be revolutionary for MeV neutrino astrophysics, because of its huge detector volume, unique event reconstruction capabilities, and excellent sensitivity to the $ u_e$ flavor. However, its backgrounds are not yet known. A major background is expected due to muon spallation of argon, which produces unstable isotopes that later beta decay. We present the first comprehensive study of MeV spallation backgrounds in argon, detailing isotope production mechanisms and decay properties, analyzing beta energy and time distributions, and proposing experimental cuts. We show that above a nominal detection threshold of 5-MeV electron energy, the most important backgrounds are --- surprisingly --- due to low-A isotopes, such as Li, Be, and B, even though high-A isotopes near argon are abundantly produced. We show that spallation backgrounds can be powerfully rejected by simple cuts, with clear paths for improvements. We compare these background rates to rates of possible MeV astrophysical neutrino signals in DUNE, including solar neutrinos (detailed in a companion paper [Capozzi et al. arXiv:1808.08232 [hep-ph]]), supernova burst neutrinos, and the diffuse supernova neutrino background. Further, to aid trigger strategies, in the Appendixes we quantify the rates of single and multiple MeV events due to spallation, radiogenic neutron capture, and other backgrounds, including through pileup. Our overall conclusion is that DUNE has high potential for MeV neutrino astrophysics, but reaching this potential requires new experimental initiatives.
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