How should fresh cherries be stored?

In the refrigerator. Do not wash the cherries prior to refrigerating, as residual moisture may promote mold growth. Store the cherries in loose plastic wrapping rather than a sealed container or airtight bag; exposure to oxygen will allow any  moisture the fruit exudes to evaporate. The cherries may keep for 3-5 days, though times may vary depending  how fresh the cherries were when you purchased them. 
 
Source(s):
GourmetSleuth.com "Food Storage Guidelines"
Utah State University "Cherries"
Bennion, M.1980. "The Science of Foods" John Wiley & Sons, NY. Pp. 201-244.