I got inspired to write this article by reading a very interesting text of @thecryptofiend about the Genus Hydra (for example Hydra viridis) and its immortality.
Why must we die?
Of course reading his article may inspire thoughts about the question if (or why not) also humans could reach immortality one day in future?
To give a small part of the answer I want to focus on one of the reasons why at present we cannot live forever:
In the nucleuses of our cells are the chromosomes (in our case 23 from our mother and 23 homologous ones from the father). The DNA of these chromosomes comprises the genes which contain the necessary information to build the polypeptides of which consist our cells (and therefore also our bodies). The ends of each chromosome consist of DNA with repetitive nucleotide sequences and associated proteins called telomeres. In case of vertebrates these sequences contain the nucleotides T-T-A-G-G-G and repeat several thousand times. Their function is to stabilize and protect the chromosomes. While an organism is getting older with each cell division the telomeres of its chromosomes become shorter and shorter until the cells are no longer able to divide. Then three scenarios may happen:
- Apoptosis: cells kill themselves whereupon proteolytic enzymes (caspases) play an important role.
- Senescence: cells stop to grow and to divide.
- If cells divide again even if they contain already shortened telomeres (like some cancer cells do) one can observe an increasing instability of the chromosomes.
At the least when the lengths of the telomeres of too many cells of a human organism fall under the critical number of less than 4 kilobases (at birth their average length is about 11 kilobases) the right moment to die has come ...
End of story? Not yet completely:
Is there any hope for immortality?
There is an enzyme named telomerase which could give us some hope. It is able to repair the telomeres at the end of the chromosomes. The problem is that firstly it is not active in all human cells and secondly its activity decreases while we are growing older. Actually studies evidence that the telomerase activity in our cells also depends on our way of life. Especially stress plays an important role and chromosomes of children who suffer under a low social status exhibit shorter telomeres than average.
But even living very healthy and being extremely rich won't make us immortal though. And I still didn't talk about the main problem: Adding telomerase to the cells slows down the shortening process of the telomeres but at the same time drastically increases the probability of cancer.
So what to do? To solve both problems Scientists interbred two kinds of genetically changed mice:
- Mice who produced telomerase reverse transcriptase (TERT), one of the components of telomerase.
- Mice "races" with the tumor suppressing genes p53, p16, and p19ARF., 
The resulting transgenic offsprings reached, depending on the different combinations of tumor suppressing genes, an up to 40 % extended lifespan.
Conclusion 1 (concerning the initial question):
It could be possible that in future tumor suppressing factors combined with an increased production or donation of telomerase may also extend the lifespan of humans as well. (Actually we are very far away from reaching 'immortality' at the moment, but I needed a term to attract as much readers as possible ... :-)
Further thoughts: Is nature (is evolution) just too stupid to create everlasting life?
No! There are examples of organisms who don't switch off telomerase when aging so that in theory they could live forever (if they don't get killed by enemies or die because of sicknesses).
Amazingly the concept to die and give ones genes to the next generation most of the time seems to be superior in evolution compared to the concept of immortality. The reason for that is that it allows to adapt more easily to a changing environment according to the motto "we age because the world changes" - also computer simulations seem to confirm that., 
From the point of view of a single individual it may be desirable to live forever but from an evolutionary perspective it is actually a big advantage that we die.
Image from pixabay: https://pixabay.com/en/cells-human-medical-biology-health-1872666/