We apply machine learning algorithms to optimize thermodynamic and elastic properties of multicomponent Fe-Cr alloys with additions of Ni, Mo, Al, W, V, and Nb. The target properties are mixing enthalpy, Youngs elastic modulus, and the ratio between shear and bulk moduli, which is often used as a phenomenological criterion for a materials ductility. We thoroughly analyze the descriptors that provide the robust performance of the machine learning models. Next, the iterative active learning method is used for the optimization of the chemical composition to simultaneously improve both thermodynamic stability and the elastic properties of Fe-Cr-based alloys. As a result, we predict compositions of thermodynamically stable alloys with improved mechanical properties, demonstrating the high potential of data-driven computational design in the field of materials for nuclear energy applications.