We present a scaling theory describing hierarchical structural organization of semidilute solutions and melts of "barbwire" bottlebrushes. Such macromolecules consist of a long flexible main chain (backbone) and several side chains emanating from each branching point of a backbone in a starlike manner. We investigate local and large scale conformational properties as well as extensional elasticity of barbwire bottlebrushes in a melt and solutions within a broad concentration range. On small length scale depending on the architectural parameters bottlebrushes exhibit either cylindrical or spherical symmetry imposed by excluded volume interactions between side chains. On a larger scale bottlebrushes behave in solution as self-avoiding chains. Upon increasing the solution concentration the excluded volume interactions get screened on progressively decreasing length scales. We identify semidilute solution regimes that differ with respect to power-law dependences of large scale and local conformational properties of the bottlebrushes on the solution concentration. The hierarchical conformational structure is translated into distinct force-extension behavior of individual bottlebrush macromolecules that can be tuned by controlling brushlike architecture. These theoretical predictions provide guidelines for molecular design of polymer materials with novel rheological and mechanical properties.