The world has become more interconnected over the past few decades. Against this backdrop, economic and financial contagion following adverse shocks can have a severe impact on the global economy. How systemic can the effects of contagion be? What specific transmission channels are involved? What is their relative importance? We address these questions using a multilayered global network model of contagion that simulates the impact of sovereign debt default on the global economy. We also develop a measure of global systemic risk and use bank stress testing techniques to quantify the systemic impact of the shock and the extent of contagion on the global economy. Our model shows that economic and financial contagion are highly non-linear, and many bystander economies can experience significant negative effects as the initial default is spread through the network. This suggests that many economies might be systemically more important than what conventional measures of size or openness might suggest.