There is a detailed review of space weather effects on humans in space by Townsend [2021]. They highlight several solar energetic particle (SEP) events that would have exceeded 30 day short-term organ damage limits from recent observations, all of which are likely to be weaker than the Carrington event. They also provide the dose limits for blood forming organs (BFOs), the heart, skin, eyes, and the central nervous system (CNS) for 30 days, 1 year, and career to give the reader a baseline reference. All these values fall in the 250--1500 mGy-Eq (milliGray-Equivalent) range for 30 days. Most of the exposure estimates for recent SEP events like that on July 14, 2000 are >1000 mGy, i.e., you'd exceed a 30 day dosage limit in a matter of a few minutes.
They examine a huge event in 775 AD that would would have caused radiation sickness and/or death in both men and women even with 40 g cm-2 of aluminum shielding (typical values used are ~5-10 g cm-2).
Update
For reference, hematopoietic syndrome (bone marrow syndrome) starts with mild symptoms at ~0.3 Gy (Gy = Gray = J/kg = mean energy imparted/deposited per unit mass) and severe symptoms occur around 0.7 Gy. Gastrointestinal syndrome begins around 6 Gy and is severe around 10 Gy. Cardiovascular/central nervous system syndrome (the most severe) begins around 20 Gy and shows full expression above 50 Gy.
Mars is only ~50% further from the Sun than Earth. The radial dependence of SEP peak intensity depends on magnetic field line connection to the source, but typically falls as $r^{-3}$ [e.g., see Lario et al., 2013]. So the peak intensity at Mars would be up to 70% weaker than that seen at Earth.
My original answer was mostly written to emphasize that while one may get some protection from the Martian atmosphere and remnant magnetic fields, this does not account for the ~6-9 month trip from Earth to Mars where the astronauts would have very little, if any, protection from particles with energies >10 MeV.
So if I am understanding the results found at https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2018SW001921 correctly, even with 30 g $cm^{-2}$ of aluminum shielding one would accumulate more than 1 mGy/day (number jumps to >100 mGy/day at 1 g $cm^{-2}$ of aluminum shielding). If we assume the lowest travel time of ~6 months (~180 days), then the accumulated dose could exceed 0.18 Gy (18 Gy) from Earth to Mars. Granted, this assumes an SEP event every day so the actual value is likely lower.
The event simulated in this paper occurred on September 10-14, 2017. It was a decent event, but not an extreme event by any means. The paper says "...one of the strongest SEP events in recent years..." which is fully accurate as there have been very few SEP events at all since the start of solar minimum in 2006-2008, i.e., this has been a very weak solar cycle. For comparison, the peak intensity of the >1 MeV protons in the September 2017 event was recorded at a few x $10^{4} \ cm^{-2} s^{-1} sr^{-1}$. The peak fluxes of ~1 MeV protons for the Bastille Eay event (i.e., July 2000) exceeded 1000 $cm^{-2} s^{-1} sr^{-1} MeV^{-1}$ or in the units of the September 2017 paper $\sim10^{9} \ cm^{-2} s^{-1} sr^{-1}$ [e.g., Tylka et al., 2001]. That is, the Bastille Eay event proton fluxes were upwards of 5 orders of magnitude larger. So even if we account for the $r^{-3}$ to Mars, a drop of 70% will not reduce 5 orders of magnitude down to the September 2017 event.
This is all based on a worst case scenario, which must be planned for because as shown above for the Bastille Eay event, one event could literally end the mission before the astronauts even arive at Mars. Further, the Bastille Eay event generated ground level enhancements (GLEs), i.e., the event was so strong that charged particles (and neutrons) made it to the Earth's surface! That is, they penetrated past both Earth's magnetic field and its atmosphere. Mars' atmosphere is roughly ~1% of the volume of Earth's atmosphere. Earth's magnetic field exceeds 65,000 nT in places while Mars' peaks out around 1500 nT, i.e., Earth's is over 40 times stronger.
The above reasons are why my original answer argued that Mars would provide little protection against a strong event like that on Bastille Eay in 2000.