As I know, nobles had a lot of chests and drawers for storages of various sort (armour, garments, etc.).
The stockage of arms and armours in armouries tend to use chests too, but baskets are seen more often. For a glimse at an armourie supplie, you can check this thesis, focusing on the accounts of the tower of London: RICHARDSON Thomas, The medieval inventories of the Tower armouries 1320-1410, submitted in accordance with the requirements for the degree of Doctor of Philosophy, University of York, department of history, 2012.
You can see the inventories of the burgundian duke for some exemples of purchasing of chests : (PROST Bernard, PROST Henri (éd.), Inventaires mobiliers et extraits des comptes des ducs de Bourgogne de la maison de Valois (1363-1477), Paris, Leroux, 1902-1913, 2 t.) (On Philippe the Bold, but others accounts come in handy.
Perhaps not your every day peasant, but if we're talking about wealthy individuals, they sure did! Henry VIII is considered to be toward the end of the "middle ages", but he kept a large portion of his immense personal wealth in chests, the most valuable of which were stored underneath the royal bed. The chests traveled with the king during his royal progresses. All courtiers and aristocrats associated with the court of Henry VIII and Elizabeth I stored their most precious belongings in chests. During royal progresses, which could occur yearly and last months, the entire court traveled with the king or queen to their various residences around the country. Due to this constant travel, everything from clothing to goblets were stored in many, many, many chests. (I'm trying to remember/find a source for this. I read so much, it's hard to remember what thing was mentioned where.)
Some chests from the 15th - 17th centuries:
A HENRY VII OR HENRY VIII IRON-BOUND CHEST
A LARGE HENRY VII OR HENRY VIII OAK AND IRON-BOUND CLAMP-FRONT CHEST LATE 15TH OR EARLY 16TH
Follow up: how often did people actually have locks on their chests?
Follow up: Where did medieval people usually keep their clothes?