Perhaps you are thinking that it should be the same as closets, which the velt is not ???? to put mezuzos on. However, the reason for that is probably because closets are less than 16 square ???? – approximately 24 – 32 square feet, or perhaps because you don’t go in and out of the closet. If they don’t meet this criteria then chances are they also require a mezuzah.
If it is a storage closet then it doesn’t need one. If it is more like a room – eg a walk-in closet, most people would put one up. Perhaps without a bracha – if you are hanging multiple mezuzos at once choose a different one to say the bracha on.
Derech: Why would you need a mezuza on the other side? The closet isn’t an entrance to the room, it’s a place that is not considered a room at all because it doesn’t have the shiur.