After decades where human spaceflight missions have been reserved to low Earth orbit, recent years have seen mission proposals and even implemented plans, e.g. with the mission Artemis I, for returning to the lunar surface. SpaceX has published over various media (e.g., its official website, conference presentations, user manual) conceptual information for its reusable Starship to enable human exploration missions to the Martian surface by the end of the decade. The technological and human challenges associated with these plans are daunting. Such a mission at that distance would require excellent system reliability and in-situ-resource utilization on a grand scale, e.g. to produce propellant. The plans contain little details however and have not yet been reviewed concerning their feasibility. In this paper we show significant technological gaps in these plans. Based on estimates and extrapolated data, a mass model as needed to fulfill SpaceX's plans could not be reproduced and the subsequent trajectory optimization showed that the current plans do not yield a return flight opportunity, due to a too large system mass. Furthermore, significant gaps exist in relevant technologies, e.g. power supply for the Martian surface. It is unlikely that these gaps can be closed until the end of the decade. We recommend several remedies, e.g. stronger international participation to distribute technology development and thus improve feasibility. Overall, with the limited information published by SpaceX about its system and mission scenario and extrapolation from us to fill information gaps, we were not able to find a feasible Mars mission scenario using Starship, even when assuming optimal conditions such as 100% recovery rate of crew consumables during flight.