The stiffness of a bag containing particles can be increased by evacuating the air inside it. This stiffness-control technology, referred to as granular jamming, has been used in force displays. In such displays, the operator feels a reaction-force sensation when deforming the bag with a hand motion. The particle-containing bag can be deformed into any shape; however, the bag cannot be converted into a thin plate. This study proposes a sheet-like mechanism that, rather than particles, contains piled-stretch fabrics inside a thin rubber bag. The sheet becomes rigid after evacuating the air from the bag, with stiffness proportional to the vacuum pressure inside the bag as observed in the granular-jamming method. The sheet thickness, which is determined using the number of piled-fabric layers, is approximately in the range of 1-5 mm. The softness and stretchability of the sheet allows it to form a spherical shape and maintain that shape after vacuuming the inner air. This study demonstrates the fundamental properties of a variable-stiffness sheet and its applications in force displays.