Eternity is a very long time, and as a biological system, my capacity would likely be reached at some point (the brain contains a limited number of neurons for example, so memory would be limited).
There's no reason why in such a technologically advanced future that excess human memory couldn't be stored on "expansion drives" of some sort. These memories would be instantly accessible in the very same way that current memories are accessible, if not more accurately; even triggered without conscious will but via association.
I think you're applying modern limitations to speculative future technologies.
Agelessness does not prevent death.
Certainly, I would wish to have my life extended to a significantly long period (a thousand years seems like a good, round number), but to exist infinitely? I could not even fathom such a concept. It would mean I would outlive the Earth, the solar system, and even the universe (assuming it has an "end").
On society and agelessness:
By the time we even reach the point of this kind of technology we will have already begun to explore population control in a polite and feasible manner. This will begin at about the time that AHCs truly hit the market, resulting in the eventual abolition of labor as we've known it. Population requirements will be vastly reduced.