We present a model to describe the resonance absorption lines of various atoms trapped in closed carbon cages. These systems may be responsible for some of the as yet unexplained diffuse interstellar bands. Model potentials for possible atom-C60 systems are obtained and used to calculate the resonance lines. The trapped atoms considered are O, N, Si, Mg, Al, Na, and S, and in all cases the resonance lines are shifted towards the red as compared to the isolated atoms. The calculated wavelengths are compared to the range of wavelengths observed for the diffuse interstellar bands, and good agreement is found for Mg and Si resonance lines. Other lines may be caused by other than resonance transitions or by trapped molecules. The oscillator strengths and the abundances are evaluated and compared with observation. Mechanisms to explain the observed band width of the lines and the existence of certain correlated pairs of lines are also discussed. Directions of further theoretical and experimental research are suggested.