Immortality is quite possible. You just have to think about it. Henrietta Lacks has an immortal body. Or, rather, her liver cells are immortal, having become a famous research cell line. But what good is that to her? (It is a great service to humanity, to be sure.)
Immortality of the soul is quite possible too, even without supernatural concepts. It goes to ontology: what is You? The memory of most people survives their death at least a little. The memory of a few people has survived 3,000 years.
For the past twenty years or so, it's been possible to preserve the details of your genome, in theory meaning we could reconstruct your body. To be sure, your DNA is just part of you, and not the most important part. But it is definitely part of you.