This is wrong - The body isn't a closed system, but an ongoing exporter of entrophy. There is no fundamental reason why "better repair mechanisms" wouldn't result in an permanent health. I don't like calling this immortality, because.. well, mishap and violence will still get you eventually, but the whole decay and slow dying thing isn't written into the laws of physics or even biology. It's just that Azathoth never had a reason to fix it.

There are animals which don't appear to age. And there have even been some successful anti aging treatments applied on animals. Even simple stuff like caloric restriction might significantly increase lifespan.

My ideal method would be cloning bodies and doing brain transplants. Of course you still need to prevent damage to the brain itself, but that solves like 90% of the other problems which occur elsewhere. And it's been shown that young blood helps the brain too. At some point we might be able to grow new brain tissue as well and keep you alive Ship of Theseus style.

Same way you prevent cuts and bruises from accumulating over the years: Repair and replacement. Your body doesn't stay whole by preventing cuts from happening, but by effectively patching over them when they do. In principle there's no reason this couldn't be applied to DNA and oxidative damage. At least for DNA we know that mutation rates vary between organisms (as indeed do rates of aging), so it's theoretically possible to lower the mutation rate.