Cucumbers are best fresh picked and eaten within the first two days of harvesting them. During that time, it's best to store them at room temperature. If you need to store them for longer than that, you can put them in the fridge, although cucumbers are subject to chilling injury at low temperatures. So, if you are going to refrigerate them straight from the grocery store, be sure to store cucumbers in the crisper bin or in the warmest part of your fridge (so not all the way in the back). You can also wrap them in plastic (English cucumbers usually are sold this way) and put them in an airtight container in the fridge to prevent chilling injury (as much as you can, anyway).