Liquids boil when their vapor pressure exceeds ambient pressure. The structure of Starship tanks limits ambient pressure to 6 bar. Therefore, to prevent boiling you must refrigerate to below -170 °C (for O2) and -145 °C (for methane).
When Starship is refueled in Earth orbit, the fuel can be sub-cooled, but not below -182 °C since that’s the freezing point of Methane. To prevent boil-off, Methane must be maintained between -182 °C and -145 °C for the duration of the voyage. The equivalent numbers for O2 are -219 °C and -170 °C. If the two tanks are in thermal equilibrium, the range is -182 °C to -170 °C to keep both liquid.
Left unregulated, are tanks going to heat up or cool down on a Mars trip? Brylle Reyes in https://www.academia.edu/934756/Thermal_Control_Handbook has an interesting concept: a standard 1 m Black Body sphere with absorptivity = 1.0, at thermal equilibrium with space. If the sphere is 1 AU from the sun, equilibrium temperature is +6 °C. At Mars distance, the temperature is -47 °C. Not surprisingly, these temperatures are close to the corresponding average planetary temperatures.
So, if we spray Starship with BBQ paint and spin it with axis perpendicular to the ecliptic, its temperature would approach these equilibrium temperatures. During the entire trip, skin temperature would decrease, but tank temperature would increase . The BBQ black Starship will need cooling to maintain tank temperatures in the “Goldilocks Zone”,.
Of course, Starship is not BBQ black. The heat shield is high absorptivity and highly insulated. The shiny side is lower absorptivity but uninsulated. It’s like a sleeping bag with one side made from a space blanket and the other side black, fluffy lamb fleece. Lay too close to a hot campfire on a very cold night, and you are in the same dilemma as Starship. If you roll the insulated side toward the fire, the insulation will protect you from overheating and you will freeze your buns on the other side. Roll over, and you will warm up again.
This is likely what Starship will do: use attitude control for passive thermoregulation to maintain tank temperatures in the Goldilocks Zone.
There are other tricks: For Black Bodies, Absorptivity = Emissivity = 1.0 at all temperatures. Real materials have differing emissivities at different temperatures. This graph shows emissivity profiles for polished SA508 steel, and with 3 different oxidation treatments. https://www.sciencedirect.com/science/article/abs/pii/S0017931017325802
Solar panels can do double duty as sunshades. Turning the nose towards the sun can warm the payload while shading the tanks.
If active-ish cooling is needed, liquid CH4 or O2 can be circulated through a radiator on the shaded side where it can radiate to the void, trying to equilibrate with the microwave background temperature. If Starship were turned nose-to-the-sun, Methane could be pumped through the bells of the Raptors as radiators. The tank contents could be cooled while the payload is warmed in the sun at the other end of Starship.