When your body replicates your DNA, the enzyme that catalyzes the reaction (DNA polymerase) cannot begin the replication at the beginning of the strand, because it has to latch on to a certain group of atoms that only exists on one side of the nucleotides. As a result, every time you replicate your DNA, you lose a nucleotide. Needless to say, this is bad news for your genes, and, by default, you. So your DNA has long strands of repeating sequences of non-coding DNA at each end called telomeres that serve as a buffer zone. However, you replicate your DNA quite a lot over the course of your lifetime, and as you get older, you begin to lose genes. The result is aging. This is speculated to be why Dolly the Sheep died around the same time as the sheep she was cloned from, even though it was already a few years old when Dolly was cloned from it. The telomeres had already been lost. This is one of the main problems for cloning, aside from the horrendous rate of failure.
The good news is that there is an enzyme called telomerase that repairs telomeres, so that you never run out of them. Cancer cells have this, which is why cancer cells are immortal. Now consider this: cancer cells are your own cells that have been mutated to not know when to stop reproducing. This means that you have the DNA to make telomerase; it is just inactive. If it was active, you would not age, because you would never lose genetic information due to replication.
This brings up an interesting point: if telomerase would keep humans from aging, and we have the DNA to make it, why is the DNA inactive? I have no way of knowing for sure, but I think the fruit of the Tree of Knowledge of Good and Evil had something that deactivated this portion of DNA when consumed. Continuing in this train of thought, might the Tree of Life have something to counter this effect? Or perhaps it has telomerase in its fruit. Of course, God could have used purely supernatural means, but He seems to like to work through the laws of nature that He created, and I see this as one way He could have done that.
Also, suppose scientists could figure out how to reactivate the genes that make telomerase? Could we make ourselves immortal? Here we start seeing the same dangerous type of thinking that caused the Tower of Babel, but I thought I’d mention it just as food for thought (though I guess that’s really what everything here is).
I just found a post that covers this in more detail, and goes beyond telomerase to other causes of age.