The approximation-theory problem to describe classes of functions in terms of the rate of approximation of these functions by polynomials, rational functions, and splines arose over 100 years ago; it still remains topical. Among many problems related to approximation, we consider the two-variable polynomial approximation problem for a function defined on the continuum of an elliptic curve in {{\mathbb{C}}^{2}} and holomorphic in its interior. The formulation of such a problem leads to the need to study the approximation of functions continuous on the continuum of the complex plane and analytic in its interior, using polynomials of Weierstrass doubly periodic functions and their derivatives.This work is devoted to the development of this area.