The transition to net-zero greenhouse gas emissions requires a rapid redesign of energy systems. However, the redesign may shift environmental impacts to other categories than climate change. To assess the sustainability of the resulting impacts, the planetary boundaries framework provides absolute limits for environmental sustainability. This study uses the planetary boundaries framework to assess net-zero sector-coupled energy system designs for absolute environmental sustainability. Considering Germany as a case study, we extend the common focus on climate change in sustainable energy system design to seven additional Earth-system processes crucial for maintaining conditions favorable to human well-being. Our assessment reveals that transitioning to net-zero greenhouse gas emissions reduces many environmental impacts but is not equivalent to sustainability, as all net-zero designs transgress at least one planetary boundary. However, the environmental impacts vary substantially between net-zero designs, highlighting that design choices exist to address transgressions of planetary boundaries.