Peel carrots and cut them in half so they are half as long. If you want to keep your carrots whole and unpeeled, that's fine. You'll just need a bowl or sealed container that can accommodate the entire bunch of carrots and some water.
Arrange the carrots in the receptacle so that there is at least 1-inch of headroom between the top of the carrots and the rim of the bowl.
Fill the bowl with clean (ideally filtered) water. Cover the bowl with plastic wrap or its sealable lid and place in the refrigerator.
Change out water once a day. Carrots will keep fresh and superbly crunchy for up to a week this way.
You can keep any root vegetable fresh and crispy this way -- try it with parsnips, beets, etc.