By the time children today are old enough to die from natural causes, we'll have a cure for that.
At some point I think we will 'cure' cellular senescence. Then humans can stay physically in their prime, indefinitely. But there are other limits, like memory. Because memories require physical space, there is only room for a limited amount.
I think we can get past that either naturally, by overwriting old memories, or artificially, by augmenting memory with technology. Research is already looking at brain augmentation with electronics.
So what will be the ultimate limit? Or will humans achieve immortality?
I'll give you that, but I think gene modification is crossing into a different realm. Can you really be called human if you have different DNA? Or perhaps its a process of rewriting the code from your original "good DNA" over your current damaged DNA with a series of tailored retroviruses? Not sure that possible, since one persons blood/fluids would probably do some serious harm to anyone else.
Regardless, if we can give cells indefinite lifespans, we can give people indefinite lifespans. No question about it. Humans are machines. If you replace the parts that fail, the machine keeps working. This raises the same questions found in Theseus's paradox. If you keep replacing parts until there's no original parts left, is it still the same machine? Or have you just made a new machine?
Physically, yes, the human body can be sustained indefinitely. But the immortality of the human mind is a question for philosophers.