I like how you are thinking. The answer to your question is… one day we might be able to BUT it might not necessarily be a good thing!
Our bodies are like big machines full of pieces and components. As bits wear out they are replaced automatically by ‘stem cells’ these produce new cells to replace the old ones. Every cell in the body has a job to do and they all have a finite life span, most cells are programmed to die off so new ones can come in. This is called Programmed cell-death (PCD) and more info can be seen here: http://en.wikipedia.org/wiki/Programmed_cell_death
If they were immortal it could cause problems because what would happen to them when they wore out or fell off? How would the body know when to replace them? In fact with PCD cells can carry on multiplying in certain parts of the body and form tumours, some of these can be cancerous.
There are plenty of treatments hopefully in the future that will use stem cells to treat illnesses/injuries but I don’t think any of them will make us immortal, but they might just allow us to live as long as a tortoise can (150-200 years). Hopefully, we’ll be able to move around a bit quicker though!
P.S. Did you know gradually every 7 years or so your cells in the body will be ENTIRELY replaced… meaning 7 years ago you were made out of a different set of 100 trillion cells!