Root Simple cites the University of California, Davis, which determined that cucumbers are sensitive to temperatures below 50°F and may develop "chilling injuries" including water-soaked areas, pitting, and accelerated decay. If you must refrigerate your cucumbers, limit it to 1-3 days and eat them as soon as possible. (We'd also suggest keeping them off the bottom shelf, which is usually the coldest part of the fridge.)
In addition, cucumbers are highly sensitive to ethylene and should be kept away from bananas, melons, and tomatoes, in particular.
Read more: Don't store your cucumbers in the fridge at Root Simple