A cross-corrugated support structure includes a sheet having a first and a second set of corrugations. The first set of corrugations is defined by a series of alternating ridges and grooves that extend the length of the sheet in a first direction. The second set of corrugations is also defined by a series of ridges and grooves that extend the length of the sheet in a second direction that intersects with the first direction. The intersection of the first and second set of corrugations creates cross-corrugations throughout the sheet. To provide compressive and tensile strengths suitable for large-scale construction applications, the sheet may be made of a carbonaceous material such as carbon fiber or graphite treated to rigidly retain a shape including the first and second set of corrugations within the sheet. The sheet may be reinforced by securing support members or additional corrugated sheets to the sheet.