The method of self-similar approximations is applied for calculating the eigenvalues of the three-dimensional spherical anharmonic oscillator. The advantage of this method is in its simplicity and high accuracy. For the case considered the authors show that based only on two terms of perturbation theory they find the spectrum with an error not worse than of the order 10-3 for the whole range of anharmonicity parameters, from zero up to infinity, and for any energy levels. The comparison with other known analytical methods proves that their method is more simple and accurate.