Fully computable a posteriori error estimates for eigenfunctions of compact self-adjoint operators in Hilbert spaces are derived. The problem of ill-conditioning of eigenfunctions in case of tight clusters and multiple eigenvalues is solved by estimating the directed distance between the spaces of exact and approximate eigenfunctions. Derived upper bounds apply to various types of eigenvalue problems, e.g. to the (generalized) matrix, Laplace, and Steklov eigenvalue problems. These bounds are suitable for arbitrary conforming approximations of eigenfunctions, and they are fully computable in terms of approximate eigenfunctions and two-sided bounds of eigenvalues. Numerical examples illustrate the efficiency of the derived error bounds for eigenfunctions.