When cucumbers are kept out the refrigerator at room temperature they should last from 3 days to a week before starting to wilt.
If you want to keep the cucumbers longer than a few days to a week then you should slice them up and place them in some water and freeze them and they'll keep longer.
Usually though you'll likely eat them well within a week or two weeks because cucumbers are best eaten fresh.
Cucumbers though do keep better when they're never refrigerated at all so if you haven't put them in the refrigerator then they should keep for over a week or two.
You don't have to refrigerate cucumbers but if you've refrigerated them then you need to continue to refrigerate them.