Is immortality achievable by humans

That would certainly depend on your definition of immortal. If you mean in a physical sense, where your body never dies, then it has not ever been documented in the last few million years of human existence. However, if you mean in the spiritual or metaphysical sense, then I suppose it might be possible, at least within human memory. People like Confucious or Shakespeare or Beethoven are remembered far beyond their own physical lifetimes. You could also say that you always live on through your children and decendants, too.

Turritopsis nutricula is the only true immortal creature