We had wild volunteer watermelons growing at the office one year and I picked them all once they got ripe because people that were not from our building were stealing them. I forgot how many there were, maybe 8 or 10. I kept them in a dark room that was cool and dry on an AV cart. AC climate controlled. I'd chill one every few days and cut it and share it with the office for the morning snack. They survived 2 or 3 weeks. Probably would have survived longer. I actually washed each with soap and water after chilling and saved the rind off of several of them to make pickles with. Wonderful! Can't recommend trying them enough.

If I were doing them at home, I might suggest washing with a 10% bleach solution and drying them, and then storing in either a spare bathroom in the tub, or in a basement, or spare bedroom on a plastic sheet just in case something happened. Maybe even on newspaper on the plastic. As long as there is plenty of ventilation. Roll them every few days.