We report on measurements of radiation hardness of plastic scintillating fibres to be used for scintillating fibre calorimetry. Fibres were irradiated by a 60Co γ-source and the effects on the emission and on the attenuation of scintillation light in a number of different fibre types were studied. Results are given on the changes in the wavelength spectrum induced by ionising radiation. The influence of optical filters, glue and surrounding gas were investigated. A Monte Carlo study is described that simulates radiation damage in a compensating lead-fibre calorimeter and predicts the impact on the performance. By combining the results of the radiation hardness measurements of the fibres and the results of the Monte Carlo, a limit is set on the acceptable dose level. If we define a contribution of 0.8% to the constant term of the electromagnetic energy resolution due to radiation damage as an acceptable performance degradation, we determine, with the best fibres that are presently commercially available, an acceptable dose limit of 7 Mrad.