A variational problem with an obstacle for a certain class of quadratic functionals is considered. Admissible vector-valued functions are assumed to satisfy the Dirichlet boundary condition, and the obstacle is a given smooth (N∈-∈1)-dimensional surface S in R ^N . The surface S is not necessarily bounded. It is proved that any minimizer u of such an obstacle problem is a partially smooth function up to the boundary of a prescribed domain. It is shown that the (n∈-∈2)-Hausdorff measure of the set of singular points is zero. Moreover, u is a weak solution of a quasilinear system with two kinds of quadratic nonlinearities in the gradient. This is proved by a local penalty method. Bibliography: 25 titles.