My understanding of the current theory of aging involves telomeres. They're a sort of genetic accessory, a cap at the end of each allele. Over time, as they are copied, the telomeres lose cohesion or wear out. The strands of DNA stop copying properly, so the cells start to break down, the organs break down, and eventually the whole system fails, i.e. death by natural causes.
Becoming immortal would mean preserving the telomeres. Some mechanism would repair any deviation from the original layout of genes, including the telomeres, so there'd be no possibility of aging or mutation or cancer. It might be an expansion of the immune system or the reproductive system.
Some implications of immortality—the body doesn't change. At all. Scars would heal over. It's possible that the brain wouldn't change either, that is, you could learn new things, but you would have difficulty breaking old patterns of thinking. Without being able to adapt to new circumstances, you'd become obsolete. The world would leave you behind, unless you and others like you tried to keep the world from changing (which would cause other problems).
If your genes don't allow any deviation, you can't reproduce. When the egg is fertilized, the immortal genes try to "correct" their nonimmortal counterpart. In effect an egg would treat sperm as a mutagen and heal itself, while sperm would erase an egg.
An accident is usually a series of mistakes or unlikely events, rather than just one. If you are alert enough, and know what to look for, you can avoid more accidents as you learn what causes them. An immortal, I believe, would develop skills to predict these patterns and avoid their effects. Over time they'd become like instincts. You could still get caught up and killed, but not as often as mortals.
Under World. It rocks!