In the laboratory study of extreme conditions of temperature and density, the exposure of matter to high intensity radiation sources has been of central importance. Here we interrogate the performance of multi-layered targets in experiments involving high intensity, hard x-ray irradiation, motivated by the advent of extremely high brightness hard x-ray sources, such as free electron lasers and 4th-generation synchrotron facilities. Intense hard x-ray beams can deliver significant energy in targets having thick x-ray transparent layers (tampers) around samples of interest, for the study of novel states of matter and materials' dynamics. Heated-state lifetimes in such targets can approach the microsecond level, regardless of radiation pulse duration, enabling the exploration of conditions of local thermal and thermodynamic equilibrium at extreme temperature in solid density matter. The thermal and mechanical response of such thick layered targets following x-ray heating, including hydrodynamic relaxation and heat flow on picosecond to millisecond timescales, is modelled using radiation hydrocode simulation, finite element analysis, and thermodynamic calculations. Assessing the potential for target survival over one or more exposures, and resistance to damage arising from heating and resulting mechanical stresses, this study doubles as an investigation into the performance of diamond-anvil high pressure cells under high x-ray fluences. Long used in conjunction with synchrotron x-ray radiation and high power optical lasers, the strong confinement afforded by such cells suggests novel applications at emerging high intensity x-ray facilities and new routes to studying thermodynamic equilibrium states of warm, very dense matter.