Place the Potatoes in a cardboard box, paper bag, mesh bag or basket to ensure good ventilation. Plastic bags won't allow them to breathe, and will shorten their shelf life considerably. Store it in a cool, humid and dark place at an ideal 45-50° temperature range. Avoid putting them in the fridge as the too cold temperature will turn the potato starch into sugar.
Rinse off Cucumbers for any dirt or grime and dry them thoroughly before storing to remove excess water on the surface which encourages spoiling. Wrap in a clean dish towel or paper towel and tuck them inside a resealable plastic bag. Put the bag into the crisper drawer of your fridge.