Sample damage by X-rays and other radiation limits the resolution of structural studies on non-repetitive and non-reproducible structures such as individual biomolecules or cells. Cooling can slow sample deterioration, but cannot eliminate damage-induced sample movement during the time needed for conventional measurements. Analyses of the dynamics of damage formation suggest that the conventional damage barrier (about 200 X-ray photons per A2 with X-rays of 12 keV energy or 1 A wavelength) may be extended at very high dose rates and very short exposure times. Here we have used computer simulations to investigate the structural information that can be recovered from the scattering of intense femtosecond X-ray pulses by single protein molecules and small assemblies. Estimations of radiation damage as a function of photon energy, pulse length, integrated pulse intensity and sample size show that experiments using very high X-ray dose rates and ultrashort exposures may provide useful structural information before radiation damage destroys the sample. We predict that such ultrashort, high-intensity X-ray pulses from free-electron lasers that are currently under development, in combination with container-free sample handling methods based on spraying techniques, will provide a new approach to structural determinations with X-rays.