This is a fun question to think about.
Copying and the Hayflick Limit
Generally what stops humans from being immortal, well at least one of the mechanisms, is whats known as the Hayflick limit. Basically its a limitation to the number of regenerations a cell line can take, before it experiences programmed death. Theres a good reason for this. The more a cell reproduces, as in copy, copy of copy, copy of copy of copy, the more 'noise' is introduced to the genome in the form of mutations, inefficiencies and ultimately, cancers. Cancer in fact largely works by disabling the hayflick limit and thus becoming immortal (Or at least until it kills the host, or the hosts immune system kills it.)
So how dow we work this into our immortal 8ft tall guys with a core?
Mother cells and Stem cells
The cores job is to store a pristine copy of the genetic blueprint of the human, as well as a reserve of stem cells that can be deployed to replace cells getting close to the hayflick limit. The stem cells are always first generation copies, because at the center of the organ is a reserve of mother cells that never die and when they divide there is always one of the divided cells that keep the original genome. But if these cells are destroyed, they will not grow back, because then the mother cells would not be original copies anymore. The mechanism that ensure the mother cells are original also ensures they can not grow back. (Note, Mother Cells are not a thing in IRL biology)
The Blood-Core Barrier
The organ features a blood-core barrier around the mother cells that only allows a heavily filtered set of blood components in, as, much like the brain which also can not reproduce its cells (mature neurons dont really have a reproductive capability), no immune system exists in there, this is to prevent auto-immune disorders attacking the core. This is not really a problem as the blood-core barrier doesn't allow any cells in except red blood cells and since red adult blood cells ALSO don't have reproductive capacity they can't be hijacked by viruses. The downside of this is a rupture or damage to the barrier makes the core extremely vunerable to infection.
Putting it together
As long as the organ is uninjured, and the mother cells are protected from infection or rupture, the core can generate stem cells forever to replace dying cells elsewhere in the body. If the core is injured however, particularly the reserve of mother cells, all bets are off, and even with close modern medical care, the patient can only reasonably expect to live another 50 or therabouts years, as now their cells will age , die, and mutate just like the rest of us.
I'm a little hazy on the exact way mother cells can create new stem cells without the mother cells themselves being prone to the hayflick limit or copying errors, but perhaps they have a unique mode generating the stem cells I guess.