Kirby Cucumber Storage
According to the New Cookbook, from Better Homes and Gardens (p. 106), pickling cucumbers should be picked and used in the same day. Standard salad cucumbers last about ten days. Trying to get from 1 to 10 is a bit of a stretch, and storage considerations can vary. 

Some people recommend the standard washing and wrapping in paper towel, others indicate plastic bagging without washing. The slime and mushiness seems to be a result of the loss of moisture through the skin. Perspiration is inhibited by waxing, this is likely the reason behind the prohibition on washing (assuming you have a waxed cuke). If you are getting the cuke from a stand it is likely fresh*er* and unwaxed.

First, ensure you are picking ideal cucumbers by verifying that the skin and flesh feels firm and without shriveling or soft spots. Then, in my experience with unwaxed cukes, you will likely get 3-5 days with them stored in a breathable container (to prevent pooling of moisture, and slime; a similar humidity issues arise from the high perspiration of grapes). Waxed, which seem unlikely to encounter, stored in a plastic bag it may last longer.

A breathable container could be one of the paperboard crates they come in (that's how I keep them), a basket, maybe a plastic perforated grape bag, or perhaps a so-called Green Bag ("breathable" bags, however absorbing ethylene is not terribly necessary here).