Immortal beings already exist. They are called "trees". Technically trees do no die because of age, although trees can get weaker over time, although I must note it depend per tree breed. Oaks in particular are known to last forever if the conditions are right. Some Holy Oaks that believers of Wotan prayed to still live today. Since the Wotan religion was driven out by Christianity approx 2000 years ago, that tells us something, eh? What is important to note is that "immortal" is not the same as "invincible", the two terms are often mixed up since gods are usually both, and some entities encountered in mythology and fantasy stories are both.
There are also a few constraints. Human memory for starters cannot be infinite... Someday it must reach its limit and how are you gonna live on then? And a lot of scientists say that even the universe as a whole must come to an end eventually, although there is no full consensus on that one yet.
Of course. Do you want to live forever? To quote Q (not THE Q) from Star Trek Voyager in the episode "Death Wish": "They haven't spoken to each other for 6 billion years. And why should they? Everything there is to be said, has been said".
In theory anything is possible, but what is realistic?