Context. NGC 2264 is a young cluster whose accretion properties can be investigated in detail by taking advantage of the FLAMES data in the context of the Gaia-ESO Survey. In fact, the analysis of the Hα emission line profile can provide us with information about the accretion and ejection activity of young stars. However, a strong nebular emission that contributes to the Hα emission can alter the profiles, with consequences for their physical interpretation. Aims. Our study is aimed at investigating the accretion and ejection properties of NGC 2264 by applying a proper treatment of the sky contribution to the Hα and forbidden emission lines (FELs; [SII] and [NII] doublets). Methods. We developed a tool, the OHαNA-method, to handle the strong nebular contribution and spectra with spurious profiles of the Hα and FELs, namely altered Hα profiles or absorption features artificially created where emission lines (FELs) are expected. We derived the quantitative measurements of relevant parameters to describe the accretion and ejection processes in young members of NGC 2264, focusing on reliable quantities derived from the width of the lines, which is relatively unaffected by the nebular emission, unlike the intensity peak, which can be altered significantly. Results. We derive the quantitative measurements related to the Hα emission line and discuss the comparison between the original and sky-subtracted spectra. We thus reveal possible profile alterations with consequences for their physical interpretation. Furthermore, we show the analysis of the variability for multi-epoch observations, also deriving the velocity of the infalling and outflowing plasma from the wings of the broad Hα emission line (in accreting stars). We also explore the mass accretion rate versus full width at zero intensity of the Hα line, namely Ṁ versus FWZI(Hα), a correlation based on the width of the emission line, which is expected to be more robust with respect to any measurement derived from the peak (e.g., Hα_10%) and possibly altered by the nebular contribution. Conclusions. We are able to ascertain that more than 20% of the confirmed accretors, which have already been identified in NGC 2264, are affected by the alteration of their line profiles due to the contribution of the nebular emission. Therefore, this is an important issue to consider when investigating accretion and ejection processes in young stellar clusters. While a small fraction of spectra can be unequivocally classified as either unaffected by nebular emission or dominated by nebular emission, the majority (> 90%) represent intermediate cases whose spectral features have to be investigated in detail to derive reliable measurements of the relevant parameters and their physical implications.