Burnout syndrome among public sector employees is a growing concern, posing a critical threat to the effectiveness, responsiveness, and resilience of public administration. Caused by a complex combination of individual and organizational stressors, burnout in the public sector may impair performance at both individual and organizational levels. This study examines the psychological well-being of civil servants working within bureaucratic systems by investigating burnout, job satisfaction, and turnover intention among a convenient sample of 437 employees of the Greek Ministry of Education and Religious Affairs—a group that, unlike frontline professionals in healthcare and education, has received significantly less scholarly attention despite its critical role in supporting core ministerial functions. Drawing on the Job Demands-Resources (JD-R) model, the study explores the relationship between key demographic variables and exhaustion, cynicism, and professional efficacy. The results reveal elevated levels of exhaustion and cynicism, alongside relatively strong professional efficacy. Burnout was negatively related to job satisfaction and positively related to turnover intention. Beyond diagnosis, the study presents innovative policy proposals to address burnout and enhance employee well-being. These include the following: redistributing workload, implementing flexible work arrangements, offering structured career-development pathways and mentorship programs, providing access to counseling, stress management and peer support, and using artificial intelligence (AI) to monitor employee well-being and automate routine administrative tasks. This study contributes to public sector innovation by applying the JD-R model to a highly centralized administrative context, offering evidence-based recommendations to enhance employee well-being and organizational performance in the public sector.