The 2D problem on an arbitrary nanohole in an infinite elastic body under remote loading is solved. It is assumed that complementary surface stress is acting at the boundary of the hole. Corresponding boundary conditions are formulated according to the generalized Young-Laplace equations. The Gurtin–Murdoch surface elasticity model is applied to take into account the surface stress effect. Based on Goursat–Kolosov complex potentials and Muskhelishvili’s technique and using conformal mapping of the outside of the hole on the outside of the circle, the solution of the problem is reduced to a singular integro-differential equation in an unknown surface stress. For a nearly circular hole, the boundary perturbation method is used that leads to successive solutions of hypersingular integral equations. In the case of elliptical and triangular holes, these equations are solved for the first-order approximation and corresponding expressions for stresses are derived in an explicit form. The influence of the surface st