In this paper we design high-order (non)local artificial boundary conditions (ABCs) which are different from those proposed by Han, H. & Bao, W. (1997 Numer. Math., 77, 347-363) for incompressible materials, and present error bounds for the finite-element approximation of the exterior Stokes equations in two dimensions. The finite-element approximation (especially its corresponding stiff matrix) becomes much simpler (sparser) when it is formulated in a bounded computational domain using the new (non)local approximate ABCs. Our error bounds indicate how the errors of the finite-element approximations depend on the mesh size, terms used in the approximate ABCs and the location of the artificial boundary. Numerical examples of the exterior Stokes equations outside a circle in the plane are presented. Numerical results demonstrate the performance of our error bounds.