The studies of structure and spectroscopy performed on radioactive nuclei during the last three decades have shown that the nuclear shell structure changes towards the drip-line and local magic numbers may appear. Doubly-magic nuclei are very rare but represent stringent tests for theories and their modelling of the nuclear interaction.In this context, we have investigated the structure and spectroscopy of the drip-line doubly-magic nucleus 24O via proton elastic and inelastic scattering (p,p'). The experiment was performed at RIKEN in the BigRIPS line, using the 24O beam produced at 263 MeV/n with RIBF with a high intensity (1780/s), and the state-of-the-art MUST2 charged particle detector. The analysis of the data gives the reconstruction of : the 24O excitation energy spectrum up to 35 MeV with the scattered proton kinematics using the missing mass method, and the angular distribution of exclusive (p,p) elastic cross section between 4 and 30 degrees c.m. via a triple coincidence nucleus-proton-nucleus. Below the two-neutron separation threshold (S2n) the statistics is too low to obtain the two excited states measured by previous experiments done at lower incident energies. Above the S2n structures are observed for the first time due to the large excitation energy range of the excitation spectra. The measurement of the excited states located at these energies would allow to test theoretical studies of low-energy dipole excitation in light neutron-rich nuclei. The statistics obtained for proton elastic scattering is sufficient to extract the exclusive (p,p) angular distributions of the 24,23,22,21O isotopes. These results constitute a new benchmark to explore proton-nucleus interaction potential features around 260 MeV/n. The comparison of elastic data set to the reaction calculations done with the microscopic reaction approach based on the G-matrix density-dependent potential indicates that this potential is suitable. However, it remains to include both the elastic and the knockout reactions in a future complete coupled channel reaction model at high energy. The root-mean-square radii of the two last bound even-even isotopes have been estimated: rm(22O)=3+-0.1 fm et rm(24O)=3.25+-0.2 fm.
