On an extended spaceflight in Starship, the liquid propellant and oxidizer will tend towards the same temperature. The tanks can be refrigerated by boiling off contents, but your mileage may vary doing that. The tanks might be warmed by rotating the ship. But you are still stuck with both tanks being the same temperature inside.
Attached is a phase diagram of oxygen and methane. There is a narrow temperature gap between solidified Methane and tank-rupturing Oxygen pressure. Of course, the boiled O2 could just be vented. But where's the fun in launching liquid O2 just to boil it off?
I can see Starship running out of Helium, cold cranking those turbines as Mars looms ahead.
These diagram graphics are from Engineering Toolbox. I’ve edited them so the horizontal axes (temperature) line up. It appears there is only a 12 °C gap between the freezing point of methane and the temperature where the vapor pressure of oxygen exceeds the safe working pressure of the Starship tank wall (6 bar).
It would be tricky to maintain tank temperature within this band using only passive means. The shiny side of Starship is uninsulated and low emissivity. The heat shield side is heavily insulated and high emissivity. I suspect that cooling the tanks would be easy since the shaded side would try equilibrate with the cosmic background radiation temperature (2.7 K). But it would be difficult to acquire heat from the sunny side due to the low emissivity of stainless steel on the shiny part of the hull, and the high insulation of the heat shield on the other side of the hull.