Leafy green storage is split into two different camps: delicate and hearty.
For delicate leafy greens (red/green leaf, spinach, arugula) they need a balanced humidity so they don't wilt or get slimy. Put them in the compostable green bags often included--they're breathability (so they can be compostable) help maintain consistent moisture levels. Plastic ziplocs also work well. Tossing in a damp paper towel helps too.
Expect at least 4 days of fresh looking greens, but probably not more than 9 days.

You can also take off individual leaves and put the base of the leaf in water and it will soak it up and rehydrate.
For the heartier greens (kale, chard, collards): simply keep in the compostable bags or another plastic bag. Due to the nutrient density and higher fiber content, they keep their shape. 
For either, if you have them on hand a bit longer than expected and they start to wilt, put them in an omelet, braise them, or cook them in a recipe that incorporates heating.