EPTAs time goes by, cells die and your body replaces them with newly divided cells. Each time a cell divides, it loses a part at the end of its DNA. Because of this, the end of a human's DNA has a section of blank or useless DNA, the telomere, to be cleaved off bit by bit with each division. Your DNA eventually shortens until it reaches a point where there's no more useless DNA, and the real, functional DNA starts to get cut off. Once this happens all kinds of problems start to accumulate and you die. There's a limit of about 70 divisions for each cell until its DNA is damaged.
Some organisms have an enzyme called telomerase that can rebuild the useless DNA, so cells can keep dividing without having its functional DNA damaged. Still, this doesn't mean that you can't get damaged in other ways. Radiation and other things can damage the DNA in any section. If the cell's DNA is damaged and it doesn't die, it could become a cancer and keep growing.
Suffice to say, you need to be some sort of geneticist with the ability manipulate DNA to find immortality. You'd then have to deal with how to repair and keep the genome healthy for hundreds of years to prevent natural diseases. On top of that, you'd have to worry about the external environment: accidents, pathogens, fast foods.
As for babies, a female person needs to have a baby before she reaches menopause. This mean she has to have a kid in her 20's-50's. Then she, and her kids, will live for hundreds of years after that.
