The Anthropocene is characterized by close interdependencies between the natural Earth system and the global human society, posing novel challenges to model development. Here we present a conceptual model describing the long-term co-evolution of natural and socio-economic subsystems of Earth. While the climate is represented via a global carbon cycle, we use economic concepts to model socio-metabolic flows of biomass and fossil fuels between nature and society. A well-being-dependent parametrization of fertility and mortality governs human population dynamics. Our analysis focuses on assessing possible asymptotic states of the Earth system for a qualitative understanding of its complex dynamics rather than quantitative predictions. Low dimension and simple equations enable a parameter-space analysis allowing us to identify preconditions of several asymptotic states and hence fates of humanity and planet. These include a sustainable co-evolution of nature and society, a global collapse and everlasting oscillations. We consider different scenarios corresponding to different socio-cultural stages of human history. The necessity of accounting for the 'human factor' in Earth system models is highlighted by the finding that carbon stocks during the past centuries evolved opposing to what would 'naturally' be expected on a planet without humans. The intensity of biomass use and the contribution of ecosystem services to human well-being are found to be crucial determinants of the asymptotic state in a (pre-industrial) biomass-only scenario without capital accumulation. The capitalistic, fossil-based scenario reveals that trajectories with fundamentally different asymptotic states might still be almost indistinguishable during even a centuries-long transient phase. Given current human population levels, our study also supports the claim that besides reducing the global demand for energy, only the extensive use of renewable energies may pave the way into a sustainable future.