Though the hut sounds promising, unfortunately it is not nearly as useful as it could be, though you do find a couple of useful items within the hut after you accept it as a reward.
After completion of the Hunting the Wild Hunt quest, if you ask the Hermit to explain the rewards again, he will mention that you can store items in the hut. This is not possible. The hut also features a fireplace which does show as targetable, but can not be used for meditation, thus also ruling it out as a safe house for meditation and combat preparation.
Owning the witcher's hut has no effect on the storyline at all.