The interrelated programs essential for cellular fitness in the face of stress are critical to understanding tumorigenesis, neurodegeneration, and aging. However, modelling the combinatorial landscape of stresses experienced by diseased cells is challenging, leaving functional relationships within the global stress response network incompletely understood. Here, we leverage genome-scale fitness screening data from 625 cancer cell lines, each representing a unique biological context, to build a network of "coessential" gene relationships centered around master regulators of the response to proteotoxic, oxidative, hypoxic, and genotoxic stress. This approach organizes the stress response into functional modules, identifies genes connecting distinct modules, and reveals mechanisms underlying cellular dependence on individual modules. As an example of the power of this approach, we discover that the previously unannotated HAPSTR (C16orf72) promotes resilience to diverse stressors as a stress-inducible regulator of the E3 ligase HUWE1. Altogether, we present a broadly applicable framework and interactive tool (http://fireworks.mendillolab.org/) to interrogate biological networks using unbiased genetic screens.