Macroscopic single-wall carbon nanotube (SWCNT) films of nanoscale thickness have significant potential for an array of applications that demand thin, transparent, conductive coatings. Using macroscopic micrometer thick polystyrene sheets as a reference, we characterize the elastic response of freestanding multifunctional SWCNT nanosheets possessing both exceptionally high Young's modulus and good durability. Thin SWCNT films (20200 nm thick) asymmetrically "doped" with dilute concentrations of superparamagnetic colloids were suspended in ethanol as freestanding nanosheets. Through repeated and controlled deformation in an external magnetic field, we measure the temporal relaxation of nanosheet curvature back to equilibrium. From the relaxation time and its dependence on nanosheet thickness and length, we extract the SWCNT nanosheet modulus through a simple viscoelastic model. Our results are consistent with nearly ideal SWCNT rigidity percolation with moduli approaching 200 GPa and limited plasticity for sufficiently thick sheets, which we attribute to the screening of van der Waals interactions by the surrounding solvent and the macroscopic nature of the deformation.