Actually, the movie Star Trek: Insurrection provides interesting insight into this very question.
In the movie, the Baku are a people who live in the "briar patch," a patch of space littered with radiation, etc. What it does for them is give them incredibly long lives having basically stopped aging. From the novelization of the movie we read:
Anij unwrapped her arms and leaned forward. "There's an unusual metaphasic radiation coming from the planet's rings. It continuously regenerates our genetic structure. You must have noticed the effects by now."
Picard gave her a small, sheepish smaile. "We've ... just begun to." He took the cup proffered by Artim, and inhaled the steam: it smelled of pomegranates and flowers. He looked up at the boy and said wryly, "I suppose you're seventy-five."
Artim blinked at him, then with charming childlike candor said, "No. I'm twelve."
The adults smiled. "The metaphasic radiation won't begin to affect him until he reaches maturity," Tournel explained. (Source)
The usual requirement for achieving the goal you seek is to declare whatever it is that grants immortality to be ineffective before physical maturity.
Yeah, but what if I want an old man?
You have two ways of thinking about this one:
The first is the Insurrection method, but rather than the immortality triggering at physical maturity, it triggers at a point of old age. The human body has several stages of development. Simply: childhood to puberty (0-12ish), puberty to physical maturity (12ish-25ish), physical maturity to middle-age (25ish-55ish), middle-age to golden years (55ish-75ish), golden years to venerable (75ish-death). There are actually more, very distinct stages, but these will suit. You can pick any one of them and simply stop the aging process when the body goes through it's chemical changes for that stage.
The second is to not really be immortal, but to age really, really slowly. This is kind of the "Elf" solution to the problem. You live thousands of years — but your aging process is linear through that period and you eventually die.
But, none of this is exactly what I want
If those solutions (or another provided by one of our other community members) doesn't suit, then you need to choose. You aren't going to find a scientifically sensible way of describing this because it doesn't exist. You need to come up with a narrative mechanism that explains the situation and allows you to move on with your story....
Rats... I just realized your question is primarily opinion-based. OK! You got lucky! You're a new user, so please take a moment to read up on what primarily opinion-based means. You might also be interested in the difference between asking for an infinite list of things (off-topic) and a finite list of things (on-topic). Cheers!