Locally grown acorn squash are available fresh picked during August and September. If you have more than you can use right away, you can store them nicely at room temperature for several weeks. For best storability, though, leave the the woody stem in place.
If you want to keep them longer, up to three months, you’ll need to keep them in a cool dark place. Just don’t refrigerate them as that is too cold. Avoid storing them near apples or avocados as these fruit release ethylene gas which shortens the storage life of your squash.