While mutation and neural degeneration partially accounts for it, it should be added that radiological biohazard is an accumulative problem - you don't get rid of rem once you've accumulated it, and after 50 rem side effects due to proteomic, neurological, and metabolic damage accrue. At present people accumulate on average 250 milirem of radiation per year.
This is a tough one.
The "simple" solution - don't accumulate radiation exposure just isn't viable.
But any attempts to repair it bump into the terminal complexity problem really quick. Biocycling would do the trick as organs, and tissues are swapped out (save for the brain). Replicators could do the same trick for the whole body at once. Inorganic cells to replace the brain and then the use of biocycling might be a work around.
If you're willing to allow the borrowing of traits from much simpler animals, genetic engineering could do the trick, but that's one I'm not sure would work.
There's a decidedly icky problem with the Replicants, though, and not one of terminal complexity (though that's the final limiting factor to the Replicant's longevity beyond mainteance of their infrastructure). The data that's stored in their brains is dependant on the physical architecture that it comes from. Just like you can't divorce digital data from its memory and its caches and its ALUs and such, you can't separate data from the synapses, and nervous tissue is the second most sensitive part of the body to biohazardous radiation. In other words, you still have to find a way to heal the trauma to the brain before you record it. While the rest of the body can be "re-written" freely, I imagine it would require some seriously high Bio, Nano, Cog, and even String to try a procedure of mass-disambiguation of abstract data (knowledge, memory, etc.) from concrete data (tissue, electrochemistry, etc.).
Another issue - proteomic damage. While we know about the genetic damage, one of the big problems we have is we don't know the process by which a gene decides which protein it can make (a singe gene could be used to make any of possibly a dozen different proteins). We simply are unaware. What we do know is radiological damage can interfere with the protein selection process, or at least appears to (systemic protein shifts have been noted in radiation cases). The question is, how to we gague the biotech level of a process that up until the codification of the human genome we were unaware even occured and even with the future codification of the proteome we still won't have a clue about?
I'm still working on conceptualizing a solution to this one - it is difficult.
Surely this is only a problem for people who are exposed to large doses of radiation in a short period of time?
The background radiation dosage won't be that much of an issue, or long-lived organisms like Redwood Trees would start to glow in the dark. One friend with a Chemistry degree who came across the Cumulative Dosage idea said that the radiation will decay as fast as it accumulates unless the subject is exposed to a large source.
What builds up isn't radiation, it's the damage caused by it. Even the smallest exposure to radiation will damage a few cells, just not enough to make a significant difference. Over time this builds up, but in most cases people don't have enough radiation damage for it to be significant in a human lifespan unless they play around with isotopes or otherwise get an abnormal exposure.
This problem then becomes one of repairing the cell damage. Given the Biotech resources available, how hard can this be?
This will be solved by Biotech 9 at the latest, since this is when the rulebook states that people become immune to radiation sickness. The challenge would be to define a way to achieve this at an earlier tech level. I'd be inclined to assume that since Biotech 9 makes someone immune in their own right, lower levels (7 or 8) would allow a stay in a hospital facility to give assisted regeneration.
Actually, this is a major problem, just not one encountered within the context of a single human lifetime. Like I say, after two hundred years under the prevailing environmental conditions of the last hundred anybody is having problems. There is actually problems with long-lived species - they either have simpler proteomes and genomes and more modular body structures or they meet the eventual end-of-life by radiation rather than by other things. Simply put, nothing that we've ever encountered ever has any actual immunity to background radiation - it's a universal slow killer. Even machines are susceptible to it.