Net-zero greenhouse gas emissions targets are driving interest in opportunities for biomass-based negative emissions and bioenergy, including from marine sources such as seaweed. Yet the biophysical and economic limits to farming seaweed at scales relevant to the global carbon budget have not been assessed in detail. We use coupled seaweed growth and technoeconomic models to estimate the costs of global seaweed production and related climate benefits, systematically testing the relative importance of model parameters. Under our most optimistic assumptions, sinking farmed seaweed to the deep sea to sequester a gigaton of CO₂ per year costs as little as $480/tCO₂ on average, while using farmed seaweed for products that avoid a gigaton of CO₂-equivalent greenhouse gas (GHG) emissions annually could return a profit of $50/tCO₂-eq. However, these costs depend on low farming costs, high seaweed yields, and assumptions that almost all carbon in seaweed is removed from the atmosphere (i.e., competition between phytoplankton and seaweed is negligible) and that seaweed products can displace products with substantial embodied non-CO₂ GHG emissions. Moreover, the gigaton-scale climate benefits we model would require farming very large areas (>90,000 km²)—a >30-fold increase in the area currently farmed. Our results therefore suggest that seaweed-based climate benefits may be feasible, but targeted research and demonstrations are needed to further reduce economic and biophysical uncertainties.