You can build regular walls, but you absolutely must install a continuous high-performance vapor barrier on the warm side of the walls and the ceiling of the storage room. The warm side of these walls is the outside of the room, not inside. Be sure to insulate the walls and the ceiling after the vapor barrier is installed. The door that opens into this room should be an insulated exterior-grade door that has an excellent threshold seal and continuous weatherstripping around all of the edges of the door. Lighting within the room is important but should not be overdone.
Light can also affect the quality and taste of wines.
You may want to install a timer switch on the storage room lights in case you close the door and leave it on by mistake. Tile flooring works great in wine storage areas because it is easy to clean in case of an unfortunate accident. Wine storage racks can be tough to construct from scratch if you are not an experienced carpenter. An inferior homemade design can put a wine collection at grave risk. The average bottle of wine weighs three pounds.