Carrots are best kept fresh in the refrigerator and inside an airtight bag like a zip-top bag, according to Food52, if you aren't planning to use them right away. The refrigerator's inherently dry air can dehydrate the carrots, so this approach combats that.
Your carrots will need to be kept in the refrigerator, but how you do it might actually matter. When kept correctly, raw carrots often remain fresh for 3 to 4 weeks in the refrigerator. If you slice or chop your carrots, you may keep them in the refrigerator for two to three weeks.
Carrot nutrition facts