The direct search for dark matter WIMP particles through their interaction with nuclei at the "neutrino floor" sensitivity, where neutrino-induced coherent scattering on nuclei starts contributing to the background, requires detectors capable of collecting exposures of the order of 1~ktonne yr free of background resulting from beta and gamma decays and cosmogenic and radiogenic neutrons. The same constraints are required for precision measurements of solar neutrinos elastically scattering on electrons. Two-phase liquid argon time projection chambers (LAr TPCs) are prime candidates for the ambitious program to explore the nature of dark matter. The large target, high scintillation light yield and good spatial resolution in all three cartesian directions concurrently allows a high precision measurement of solar neutrino fluxes. We studied the cosmogenic and radiogenic backgrounds affecting solar neutrino detection in a 300 tonne (100 tonne fiducial) LAr TPC operating at LNGS depth (3,800 meters of water equivalent). Such a detector could measure the CNO neutrino rate with 5 sigma sensitivity, and significantly improve the precision of the 7Be and pep neutrino rates compared to the currently available results from the Borexino organic liquid scintillator detector. Measurements with ~2%, ~10% and ~15% precision for 7Be, pep, and CNO neutrinos, respectively, are possible.