If you are just worried about ionizing radiation then there may be biological solutions ...
1) Build a radiation-proof skin/shell. Possibly impractical, as radiation in space can be fairly penetrating - even a few mm of aluminium won't keep it all out. But you could develop a species with a very thick skin (inter-locking plates to maintain flexibility like an echidna or armadillo?) full of heavy metals. Would be ugly, heavy and lacking in mobility depending on how much shielding you required. Or could just limit the shielding to the reproductive system in which case people might have short cancer-ridden lives but would still reproduce reliably.
2) Reduce the effects of genetic errors. For example - upgrade the immune system to actively fight all likely types of cancer, or add nano-bots to continuously scour the body of tumors and other radiation-induced irregularities.
3) More high tech - harden the genetic code so it is more immune to errors resulting from ionizing radiation. This might be done a number (or combination of ways)
- Add a lot of error-correction. For example, you could duplicate or even triplicate the number of copies of each gene within each nucleus. Then when generating new proteins etc from the genetic code each cell would check if all copies of the code were identical. If so, it would 'run' the genetic code, if not it would discard its own nucleus and get a 'copy' from a neighboring cell. Continuous 'data-checking' to test that your multiple copies were in agreement. Obviously requires significant re-writing of the DNA/RNA manipulation chemistry in cells.
- Harden the DNA/RNA coding. For example, move from 3 codons to 4 or even 5 codons in the 'table' that relates codon ordering to amino acid placement in proteins. Probably an even more significant change in biochemistry.
- Harden the chemistry of genetics itself - replace the coding mechanism by some more robust chain molecule/system that can still carry genetic info. Siliconized sugar back-boned DNA, multiply cross-linked 2-D genetic materials, storage to nanoscopic mineral structures, whatever...
Previous generalist answer below:
You almost certainly mostly mechanical.
Life is very adaptable to extremophile adaptation (very hot/cold, highly acidic, heavy metal poisons, radiation, low nutrient availability) but even after millions of years of evolution, the organisms that can live in the most extreme settings are inevitably very simple organisms. For example, there are plenty of lichens in Antarctica but no trees or higher plants, and no permanent resident vertebrates. Bacteria adapt to hot springs, but you won't find fish living in them. There appear to be physiological limits to natural adaption of mammals to very high altitudes.
So far more practical than trying to bio-engineer a complex air-breathing organism, with a resource-hungry brain, to survive in space - it would likely be much simpler to either:
- take just the important 'parts' (brain, some glands, some internal organs) and install them in a mechanical 'body' designed with its own radiation shielding, reactor, long-term food supply (or microbial bio-farm that recycles your waste products into food), thrusters, grabbing arms, radar and radio. Or,
- go the whole hog and download personality into an entirely mechanical 'body'.