After picking cucumbers the stay fresh for around 5 to 7 days after picking the cucumbers off the vine.
Some people recommend storing cucumbers in the refrigerator to make them last longer which you can do but I find leaving the cucumbers out on the counter at room temperatures makes them last longer because they don't suffer cold injury.
I've kept cucumbers out on the counter for a month or two and they've still been good to eat although they were not as fresh.
So you can refrigerate cucumbers if you prefer but leaving them out at room temperatures works best for me.