Store in crisper drawers in plastic bags lined with paper towel.
If your carrots don’t have their greens removed already, remove before storing. The greens will draw moisture out of the carrots.
They can also be stored in the refrigerator in a container of water.
The water keeps them plump and crispy but will need to be changed out every few days.
For the long term, blanche and freeze. Blanching and freezing may seem like a lot of extra work but the improved texture is worth it. Don’t bother with peeling the carrots. So many nutrients are in and just under the skin. Give them a good scrub with a brush.