The shiny silicon body does not need oxygen, food, water, or large crew quarters.
This means the same form-factor starship (Effective length, width, and mass) can have a very beefy superstructure, and lots of heavy shielding (all the mass you would devote to having food, water, sanitation, and atmosphere reprocessing can all be shielding instead.) Give it enough hydrogen-rich shielding material, and even
cosmic "OH MY GOD" particles can be effectively prevented from reaching the main computer, where your consciousness lives.
At that point, it is "Do I have enough fuel to power my data core that long, will my ship's exterior suffer from
radiation embrittlement, and will I go crazy from deep space with nothing to do for centuries at a time?" rather than the normal questions.
As for radiation hardened electronics, we DO in fact know how to make these. The deal is that they cannot be made to also be "High speed", or "Low energy/High Efficiency." They have very thick layers in the lithography process, have many redundant drainage gates and redundant standby logic gates in the design, and several other things. They are indeed NOT magic bullets though, and DO fail under sustained, heavy radiation exposure. (So, prevent the heavy exposure.) Combine radiation hardened electronics, with a slow processing speed, and VERY aggressive shielding, and you have the same apparent effect to the consciousness as time dialation, as seen from point of view of that consciousness. (At high velocities, every particle you encounter on your way to destination is effectively a direct impact with a high energy cosmic ray. Remember that. To try anything at very high speeds, you need a redunculous amount of forward facing shielding. Either "Slow, with slow hardware, and lots of shielding" or "Very fast on organic hardware, with lots of shielding" still requires lots of shielding, and in both cases, it seems as if less time has passed than has actually passed.)