Tasty 3-5" cukes great for eating fresh or pickling.
Store cukes dry, wrapped in plastic & in the warmest part of your fridge, which is likely your door. Odd, we know but cukes are sensitive to temps below 50 degrees.
You can even store on your counter but make sure they're not close to your high ethelyne producing fruits like pineapple, apple, grapes, mango, melons, bananas & tomatoes.
If you just can't deal with all that, put 'em in a bag & throw 'em in the crisper. Just don't forget about them! And don't forget to wash!