A modeling study of the potential for storing energy in the elastic deformation of springs comprised of carbon nanotubes (CNTs) is presented. Analytic models were generated to estimate the ideal achievable energy density in CNTs subject to axial tension, compression, bending and torsion, taking into account limiting mechanisms such as the strength of individual CNTs, the onset of buckling, and the packing density limitations of CNT groupings. The stored energy density in CNT springs is predicted to be highest under tensile loading, with maximum values more than three orders of magnitude greater than the energy density of steel springs, and approximately eight times greater than the energy density of lithium-ion batteries. Densely packed bundles of precisely aligned, small diameter single-walled carbon nanotubes are identified as the best structure for high performance springs. The conceptual design and modeling of a portable electric power source that stores energy in a CNT spring are presented as tools for studying the potential performance of a system for generating electricity from the CNTs' stored mechanical energy.