Most costumes are stored in costume houses. Each costume house is a little different, but by and large, they are storing costumes in the same way that you would store any clothing - just on a much larger scale. Clothing is stored on hanging racks, hats are stored on shelves or hang on pegs, jewelry is often stored in drawers or hung up, etc.
Angels Costumier is the largest costume house in the world. They have over eight miles of hanging racks arranged chronologically in order to make finding costumes for the proper period easier.
I would highly suggest visiting their website for more information. They also offer a tour called “Behind the Seams” if you would like to learn more. You can also check out this interview I did with Angels about a year ago.
There are literally hundreds of costume houses around the world that provide specialty items for film and television, but if you wish to learn more, some of the best-known places are:
Email Submissions: submissions@recycledmoviecostumes.com
Follow: Website | Twitter | Facebook | Pinterest
Note: Images Copyright of Angels Costumier