The rapidly growing theories of networks and complex systems have been recently adopted to interpret the efficiency and robustness of various economic markets. Based on these theoretical underpinnings, the present paper describes a structural input-output analysis of the inter-sectoral linkages and main activity clusters of the Greek economy, which is modeled as a complex network. Such an analysis employs suitable network metrics to measure the centrality and influence of each sector-agent on the other ones, and the possibilities for clustering of related (groups of) activities. Key sectors related to the production of tradable goods and services are identified, in terms of their marginal ability to pull the total economic activity. Critical sectors are also determined in terms of their ability to retain the interconnectivity and strengthen the stability of the whole economic system. It is argued that more synergies within and among the activity clusters, through the creation of integrated value chains, would allow better coordination of policies, more efficient allocation of resources and enhanced diffusion of knowledge.