Closets are, by design, places for storage. Most rooms will have a closet (or pantry). Some rooms have multiple closets. And closets come in a variety of shapes and sizes. Larger ones give you the room to refit the closet for some other purpose, should you find you do not need the storage space. Closets over and under stairs offer some of the best opportunities for being refitted.
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Things to do with a Closet
Clearly, a closet over basement stairs was most likely designed to serve as either an extra pantry or for storage of household items, such as brooms, mops, buckets and the like. If the closet is in a dark corner of the house and stays relatively cool, you could also use it for a sort of root cellar. Or for a drying room if you grow your own herbs. Garlic and other herbs could be strung up to dry here, making them ready for storage and later use in cooking.
Perhaps you would like to convert the closet to something else. Even though we live in the digital age, many people still take film photographs. If film photography seems like a hobby you could enjoy, you could convert the closet into a dark room. Perhaps turn your photographs into art that you sell or show.
Another use for a closet is to turn it into a small wine cellar. Build in wooden racks for holding bottles of wine. If you wanted to keep any refrigerated, have an electrical outlet run into the closet and you can add a small wine refrigerator. The closet works well because it keeps your wine out of sunlight.
You can also remove the door to the closet and refinish the frame, creating an open nook out of the space. Remove any shelves. Refinish the walls and floor if necessary. Now you have a small, extra space attached to the adjoining room. In this space you can install all sorts of built-in furniture. For instance, you might put in a small desk and turn the space into a mini-office. You could install bookshelves and make the space a recessed library. Consider what you need in your house and create the space to suit your purposes.