We study the problem of the self-assembly of nanoparticles (NPs) into finite mesoscopic structures with a programmed local morphology and complex overall shape. Our proposed building blocks are NPs that are directionally functionalized with DNA. The combination of directionality and selectivity of interactions allows one to avoid unwanted metastable configurations, which have been shown to lead to slow self-assembly kinetics even in much simpler systems. With numerical simulations, we show that a variety of target mesoscopic objects can be designed and self-assembled in near perfect yield. They include cubes, pyramids, boxes, and even an Empire State Building model. We summarize our findings with a set of design strategies that leads to the successful self-assembly of a wide range of mesostructures.