The inner walls of mitochondria, cabbage leaves, and even the Himalayas are all examples of thin sheets: objects with a thickness much smaller than their length and width. Despite their differences in size and in material composition, similar patterns emerge when sheets are crumpled or forced into a small three-dimensional space. As the compaction progresses, the deformations focus into increasingly sharper features that look like the network of peaks and creases found on the surface of a balled up piece of paper. In this regime, external forces are straining the membrane, causing the elastic energy to localize while leaving most of the surface smooth and undeformed. In addition to forming a collection of deformations and smooth facets, different sections of the sheet come in contact and thus, a description of the interior configuration of the packed sheet is necessary for a complete understanding of the final structure and how it can resist further compression while maintaining a low volume fraction. With the experiments described in this thesis, we investigate the role of mechanics and geometry in highly confined sheets by exploring and quantifying elements of their inner design. We carry out experiments on sheets composed of materials with varying elasticity: Aluminum and Polydimethylsiloxane (PDMS), and probe their configurations using X-ray tomography and optical imaging techniques. Aluminum sheets are confined within a sphere and X-ray tomography aids in imaging their interior structure. In this case, the local metrics of the geometry are found to be largely isotropic and homogenous, absent of the method of preparation. However, the films spontaneously form localized order where separate sections align into stacks of facets. In the second set of experiments, we make thin PDMS membranes and confine them within a cylindrical geometry where we can contrast reducing the volume with either the flat or curved boundaries. For this set-up, optical imaging permits dynamical measurements over a much larger range of volume fractions. We are able to induce global orientational order in these membranes with high reversibility. Interestingly, even at low densities, both types of sheets form localized stacks of facets independently of the boundary conditions. This spontaneous, localized order emerges as a defining feature of membranes packed into a small volume and is a likely contributor to their structural rigidity.