That the telomere doesn't shorten basically means the DNA doesn't "age"; cells don't become more likely to mutate after X duplications begin to damage the code, like ours do.
No, it's not. DNA transcription errors occur all the time in our bodies, but there are elaborate repair mechanisms in the DNA that either correct it or, if its uncorrectable, kill off the cell.
But, sometimes these mechanisms fail and a cell survives with mutated DNA. If the right genes are mutated, then it grows uncontrolled and you've got cancer.
In order to get immortality, these repair mechanisms would have to be as robust after 1000 years as they are after 20 years. In humans, the repair mechanisms tend to degrade as you age, which is why, the theory goes, people are more likely to get cancer as they get older. It's not that there are more errors then, but that they're not corrected as well.
That the telomere doesn't shorten basically means the DNA doesn't "age"; cells don't become more likely to mutate after X duplications begin to damage the code, like ours do.