The stability analysis of socioeconomic systems has been centered on answering whether small perturbations when a system is in a given quantitative state will push the system permanently to a different quantitative state. However, typically the quantitative state of socioeconomic systems is subject to constant change. Therefore, a key stability question that has been under-investigated is how strong the conditions of a system itself can change before the system moves to a qualitatively different behavior, i.e., how structurally stable the systems is. Here, we introduce a framework to investigate the structural stability of socioeconomic systems formed by the network of interactions among agents competing for resources. We measure the structural stability of the system as the range of conditions in the distribution and availability of resources compatible with the qualitative behavior in which all the constituent agents can be self-sustained across time. To illustrate our framework, we study an empirical representation of the global socioeconomic system formed by countries sharing and competing for multinational companies used as proxy for resources. We demonstrate that the structural stability of the system is inversely associated with the level of competition and the level of heterogeneity in the distribution of resources. Importantly, we show that the qualitative behavior of the observed global socioeconomic system is highly sensitive to changes in the distribution of resources. We believe this work provides a methodological basis to develop sustainable strategies for socioeconomic systems subject to constantly changing conditions.