Full on Highlander style immortality is impossible, sure. You can't make people bullet-proof or getting-hit-by-a-train-proof. But what theoretically could be possible is curing all terminal diseases and developing gene therapies that drastically slow down the aging process, thereby significantly raising the average life expectancy. It might be impossible to make people live for thousands of years, but it might not be impossible to make them live for hundreds.

I saw a TED talk once with a genetic researcher (I think) who was talking about the idea of "curing old age." He brought up the overpopulation issue as that seems to be the most common objection to this idea, and I think he made a pretty good point about it. To very roughly paraphrase he basically said that if such an option were available, it would then be up to humanity to decide whether they want to keep breeding and dying at the pace they do now, or whether they want to live much, much longer but breed less. His point was that we shouldn't make that decision for the future people who would benefit from it, we should let them decide that for themselves.

Here's the video for anybody interested. It's been several months since I've seen it and I didn't watch it again when I found the link, so sorry if anything I said about is wrong.