Though it is conceivable to extend our lifespan indefinitely with sufficiently advanced technology, as more time pass, it becomes inevitable that something will eventually kill you, as Fluttershutter noted. If an indefinite extension of your life is what you seek, then your answer lies within the domain of the empirical sciences. But indefinite is not the same as infinite. Ultimately, attempting to achieve immortality in the corporeal domain is futile, as everything that exists within it are contingent; the very fact that we exist in a domain of becoming, of change, death is inevitable.
Even as biological entities, we already face the matter of the Ship of Theseus, as the cells in our body gets replaced very regularly. I forgot what the length is, but after an interval of some timespan, every cell in your body gets replaced; strictly physically, you may be a completely different person from when you were born. However, it would be erroneous to say that the you of today and the you from when you were born are disconnected entities; you are definitely a continuation of the you from when you were an infant.
But while I consider the pursuit of immortality to be futile in the physical domain, it is a very different matter when one considers the metaphysical domain; to achieve true immortality, we must fundamentally transcend the realm of the corporeal domain. We, in the corporeal domain, are manifestations of higher principles. Here is a question. Consider the idea of the number "7", and consider a set of seven items. Is either one more real than the other?
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Perhaps if one is strictly concerned with the corporeal domain, the set of seven items may be considered more real with the idea of the number 7, because the former physically exists, whereas the latter is just an idea that has no physical properties. But, I think the idea of the number 7 is more real than the set of seven items, because the latter as a coherent whole is contingent on the idea of the number 7; put in another way, without the idea of the number 7, we would not even recognize the set of seven items as being united by the concept of 7 in the first place -- they would simply be seen as some random collection of items. So, while the idea of the number 7 itself does not physically exist, it is the principle by which all other set of seven physical items exists; while any set of seven physical items physically exists, they do not derive their source of reality (specifically as a set of seven items) from themselves, but from a higher principle (i.e. the idea of the number 7).
And so, I think that the idea of Iforgotmybrain, so to speak, is more real than the particular corporeal manifestation of Iforgotmybrain as a flesh and blood human being; no matter how the physical person of Iforgotmybrain may change through the course of his life, he is still recognized as Iforgotmybrain instead of some random aggregation of cells, because the idea of Iforgotmybrain has many contingent manifestations. To avoid potential misunderstanding, while I say that, for example, the idea of Iforgotmybrain is more real than the corporeal manifestation of Iforgotmybrain, this does not imply that the real Iforgotmybrain as a whole is him as an idea, and that his corporeal manifestation is somehow "fake". Wholistically, Iforgotmybrain is both the idea and the corporeal manifestation; Iforgotmybrain exists simultaneously in multiple levels of reality, of which corporeal reality is the lowest level, and is the one in which we obviously can most readily sense.
To return to the question of immortality, we see here that there is relatively greater stability of the idea of Iforgotmybrain compared to Iforgotmybrain as a corporeal manifestation; the latter may go through many changes throughout time, but are all united by the idea of Iforgotmybrain which is fixed. This may be speaking roughly, but because the idea of Iforgotmybrain is not subject to change, hence not confined to the domain of becoming, it does not die (for dying is a type of change or becoming), and so it is immortal. And, because the idea of you is you who exist in a higher domain, that is where the source of your immortality lies (and in a domain where change does not happen, time does not exist as a limiting condition -- it is beyond time).
I could have spoke more precisely, but I hope I have sufficiently conveyed what I intended to convey. I say "idea", but, really, to say "essence", "soul" or "principle" would be more appropriate. Such things in themselves do not have physical manifestations much like the idea of the number 7. But, that does not mean that they do not exist; they do exist, but in higher domains that inherently cannot be sensed empirically, but can be understood (at least to an extent) intellectually.
While we can talk about extending the life of your corporeal body indefinitely with the use of science, immortality itself would require you to look beyond the preservation of your physical body.