I don't think it's biologically impossible to achieve immortality - just beyond our current knowledge. After all, for the most part our bodies are just biological machines; so it seems likely that, as our understanding of them increases, we will reach a point where we can easily repair and rejuvenate them. Another possibility would be uploading our consciousnesses (whatever they are) to the net - and let's face it, most of us already spend much of our time here anyway.

Of course you can then make the case that we might never/not any time soon reach either of those stages.