We present an approach to estimate the elastic constants of molecules and nanoparticles, based on the analysis of thermal fluctuations from Monte Carlo (MC) or molecular-dynamics (MD) atomistic simulations. The method and the force-field used for these calculations have been tested by the calculation of Young's modulus of a graphite sample along the basal plane; the calculated value was found to be 1.07 TPa, in very good agreement with the experimentally determined one of 1.02 TPa. The results on a carbon-based nanotube indicate that for the longitudinal direction of the particle, the value of the elastic constant is on the order of 400 GPa. The elastic constant of the considered nanotube in the radial direction is significantly lower, the predicted values being in the range 4-7 GPa. The method was also applied to the elastic constants of a type of siloxane-based nanostructure, whose longitudinal elastic constant (30 GPa) is an order of magnitude lower than the corresponding value for the carbon-based nanotube.