In this work we describe a detailed study of the conformational properties of flexible polymer chains confined into rigid wall spherical cavities of different sizes, with dimensions comparable to those of the chains. An off-lattice Monte Carlo algorithm has been used to simulate the behavior of excluded-volume, freely jointed chains in a wide range of concentrations. The considered properties include the size and shape of the chains, different densities of the system, and the orientation of certain characteristic vectors. All these properties are analyzed as a function of the position in the spherical cavity, trying to determine how each property is affected by the presence of the spherical rigid wall. A comparison with the results found in the literature for polymer systems confined between infinite parallel plates is done whenever possible, with some significant differences.