The Job Demands-Resources (JD-R) model is an integrative theoretical framework for monitoring workplaces with the aim to increase job engagement and prevent burnout. This framework is of great interest since the management of job resources and demands can negatively affect employees, especially in organisational contexts characterised by high job demands. This study uses the job demands-resources model to investigate the relationships between organisational climate, role stress, and employee well-being (burnout and job satisfaction) in public organisations. This is a descriptive, cross-sectional study. The research participants are 442 public employees. A structural equation model was developed (organisational climate, job satisfaction, burnout, role stress). These confirm that organisational climate is correlated with role stress (−0.594), job satisfaction (0.746), and burnout (−0.408), while role stress is correlated with burnout (0.953) and job satisfaction (−0.685). Finally, there is a correlation between burnout and job satisfaction that is negative and significant (−0.664). The study confirms that a positive organisational climate could lead to less stressed and burned-out workers and, at the same time, to more satisfied employees with improved well-being.