This contribution provides a conceptual analysis and a quantitative comparative assessment of three technology chains that enable a carbon neutral chemical industry in a net-zero-CO2 world. These are based (i) on the use of fossil fuels and current chemical processes and infrastructure coupled with carbon capture and storage (CCS route), (ii) on the use of captured CO2 as a feedstock together with "green" hydrogen in new chemical processes (CCU route), (iii) on the use of biomass grown and processed for the specific purpose of making chemicals (BIO route). All routes are feasible and have different pros and cons. Such pros and cons are first discussed through a qualitative comparison of the three routes for a generic chemical product, and are then quantitatively assessed for the specific case of methanol production. In this case, the CCU route results in an electricity consumption 10 to 25 times higher than that of the CCS and BIO routes (excluding the electricity required for heat production), mostly due to the electricity required to produce hydrogen. At the same time, the BIO route requires a land capacity about 40 and 400 times higher than that required by the CCU and CCS routes, respectively. Furthermore, when considering a net-positive-CO2 emissions world, the CO2 emissions of the CCU route grow about 8 to 10 times faster than that of the CCS and BIO routes. On the one hand, we identify key hurdles in all cases. These are (i) the availability, accessibility, and acceptance of CO2 storage sites for the CCS route, together with the continued use of fossil fuels; (ii) the very high electricity and energy demand for the CCU route, with the associated strict requirement of very low carbon-intensity of the electricity mix; (iii) the very high availability of land for biomass growth in the case of the BIO route, with the associated risks of conflict with other uses. On the other hand, we underline that the CCS route offers the possibility of using existing technologies and infrastructures, without the need of a complete reshaping of the chemical industry, and of permanently removing CO2 from the atmosphere, hence representing a key element not only in the netzero-CO2 emissions world studied here, but also in a net-negative-CO2 emissions world.