Try one of these three tips:
1. Place your green onions in a jar, and fill with an inch or two of water (just enough to cover the roots). Then, place the jar on the windowsill in your kitchen. Your onions will not only stay fresh, but continue to grow. Change or add water every couple days, as needed.
2. Place your green onions in a jar with a bit of water (just like #1). Then, cover them with a plastic bag, and store them in the refrigerator. Replace the water every couple days.
3. Wrap your green onions in a slightly damp paper towel. Then, place them inside a plastic bag or storage container. Remoisten the paper towel, if it dries out; replace it, if it because too wet.