Ecological diversity is ubiquitous despite the restrictions imposed by competitive exclusion and apparent competition. To explain the observed richness of species in a given habitat, food web theory has explored nonlinear functional responses, self-interaction or spatial structure and dispersal - model ingredients that have proven to promote stability and diversity. We here instead return to classical Lotka-Volterra equations, where species-species interaction is characterized by a simple product and spatial restrictions are ignored. We quantify how this idealization imposes constraints on coexistence and diversity for many species. To this end, we introduce the concept of free and controlled species and use this to demonstrate how stable food webs can be constructed by sequential addition of species. When we augment the resulting network by additional weak interactions we are able to show that it is possible to construct large food webs of arbitrary connectivity. Our model thus serves as a formal starting point for the study of sustainable interaction patterns between species.