I had heard of this before yet it seems pretty impossible to me as normal lifespan of human being nowadays is cca 70-80 years (much improved from those 20-30 we started off with) and biological expectancy of human body (i.e. how long can human body possibly work in ideal conditions) is around 200 years.
The theory is that we come up with technologies, chiefly nanotechnology, that is capable of safely repairing our body. If we do that then biological immorality is essentially guaranteed, of course it wouldn't protect from physical trauma. The only thing that will stop us from accomplishing this is if our bodies aren't compatible with such machines, or if legislation is passed against it.
Think about it this way, assuming evolution from a single cell is what occurred. Undesigned, unintentional evolution, through mutations, created organisms that are able to grow, differentiate and repair themselves for decades (some for hundreds of years). Look at how complex we are, how our wounds heal, how we grow from little fetuses into grown adults simply through eating foods, drinking liquids and breathing air. Now consider that we have the gift of intention, intellect, etc, so as we understand biology and technology more, we can do better than what happened only through chance in nature. Nature never made a car, we did. So is there any reason to believe we can't create something better than nature did? As in, find technology that is capable of repairing and refreshing human cells.
Keep in mind, that even if this all comes true, the average person, in our current environment, would only live for a few thousand years. Deaths from accidents, murders, etc, would put our lifespan far below anything that approaches immortality. There would have to be big cultural changes or a huge amount of luck to overcome that.