The DEAP-3600 Dark Matter detector, located 2 km underground at SNOLAB, uses up to 3600 kg of natural atmospheric argon as its detection medium. The isotopes 36Ar and 39Ar are present in small quantities in argon found in the atmosphere, and so are present within DEAP-3600. These isotopes can be examined in data collected for a Dark Matter search. 36Ar could undergo an exotic decay process called neutrinoless double electron capture. This decay violates lepton number conservation and its observation would provide a clear indication of physics beyond the Standard Model of Particle Physics. Three modes of this decay with different final states could be detected with DEAP-3600 if they occur. A search for this decay was performed on an isotopic exposure of 6.67 kg-years with the DEAP-3600 detector. A fit to this data has yielded lower limits on the half-life of 36Ar of T > 4.78e20 years. 39Ar is a beta-emitting isotope and is a major background for DEAP-3600 and other argon-based dark matter experiments. A measurement of the specific activity of 39Ar in natural atmospheric argon has been performed on a total argon mass exposure of 2200 kg-years taken over a period of 12 months; 185 individual measurements were made. The specific activity is determined to be (0.953 +/- 0.028)Bq/kg, corresponding to a concentration of (7.6 +/- 0.2)e-16 g(39Ar)/g(atmAr).