Yes. And most server hardware is already at least ECC ram. You may still want some light radiation shielding to prevent the worst, maybe some heavier shielding for solar flares. But beyond that, simple error correction can be baked into the software - ecc the bootloader and filesystem and you are mostly good to go

Isn't ECC only applied to RAM? Do any current gen CPUs have ECC cache?

I can't comment on AMD or Intel but Apple Silicon definitely uses ECC for at least the system level cache. On top of that it performs cache healing (swapping out bad lines for spares) on every cache level every time the system boots.