We introduce a two-dimensional continuum deposition model of spatially extended objects, with an effective repulsive contact interaction between them represented by a parameter 0<~q<~1. For q=0, the deposited network is uniformly random, while for q=1 particles are not allowed to overlap. For 0<~q<1, we carry out extensive simulations on fibers, needles, and disks to study the dependence of the percolation threshold on q. We derive expressions for the threshold near q=0 and q=1 and find good qualitative agreement with the simulations. The deposited networks produced by the model display nontrivial density correlations near percolation threshold. These are reflected in the appropriate spatial correlation functions. We study such functions close to q=1 and derive an approximate expression for the pair distribution function.