We consider finite element approximations for one dimensional boundary value problems of semilinear elliptic stochastic partial differential equations (SPDEs) driven by a fractional Brownian motion with Hurst parameter $H<1/2$ based on the well-posedness of the problem. We make use of a sequence of approximate solutions with the fractional Brownian noise replaced by its piecewise constant discretization to construct the finite element approximations of the SPDEs. The error estimate of the approximations is derived through rigorous convergence analysis.