I keep my shavings in a large construction trash bags. I also keep my bales of alfalfa in the either 45 or 55 gal heavy duty construction trash bags. It works for me.
Just stick it somewhere dry. If you have a small amount inside any type of container works fine. If you have a large amount a pile preferably inside a building and covered with a tarp. Also make sure it's fenced off from the animals or they'll dig out the pile and use it up the shavings early.
You don't need as many pellets as shavings. Where I have to put 3-4" of shavings in a horse stall I only need a thin coating from 3-4bags of pellets. You can prewater it a little or just wait till it absorbs moisture and a single layer barely covering the floor will turn into many inches of sawdust. It absorbs a ton of water and moisture. Put a handful of pellets in a large cup of water and you'll have a dry cup of sawdust by morning. Put in a handful of sawdust and you'll still have water with some soaked sawdust in it. Downside is pellets are much more expensive than bulk shavings and you have to handle individual fairly heavy bags for their size. I unload bags in the 100s every spring and fall whereas when we did bulk sawdust we'd have them fill an old grain wagon and just leave the sawdust in the wagon. Shovel it out to the stalls as needed. Alot less handling and no broken bags. There's always a dozen or so broken bags every year.