In this paper we present error estimates for the finite element approximation of linear elliptic problems in unbounded domains that are outside an obstacle and a semi-infinite strip in the plane. The finite element approximation is formulated on a bounded domain using a nonlocal approximate artificial boundary condition. In fact there is a family of approximate boundary con-ditions with increasing accuracy (and computational cost) for a given artificial boundary. Our error estimates are based on the mesh size, the terms used in the approximate artificial boundary condi-tion, and the location of the artificial boundary. Numerical examples for Poisson's problem outside a circle and in a semi-infinite strip are presented. Numerical results demonstrate the performance of our error estimates.