Just by looking, it's easy to see that Dragon Ball's Master Roshi has been alive for a long time. However, despite his aged demeanor, his appearance only tells half the story. In fact, the Turtle Hermit, who lived through the first battle against King Piccolo, is well over 300 years old. While the Dragon Ball universe is home to beings who have lived for thousands or even millions of years, few humans approach anywhere near Roshi's age. This raises an important question: is it possible that Master Roshi has actually obtained immortality?
The Immortal Phoenix
Goku first meets Master Roshi after helping bring Turtle back to the ocean. Although Roshi eventually gives Goku the Flying Nimbus as thanks for his altruism, he first offers the Immortal Phoenix in the manga and Japanese anime. Phoenixes are well known for their properties of eternal life. In the world of Dragon Ball, a side comment from the Hermit implies that they can even bestow immortality.
Unfortunately, Turtle informs them that the Immortal Phoenix ironically died of food poisoning the year prior. While Roshi could have theoretically stocked up on Phoenix Downs during the bird's life, the ease with which it passed away makes its powers seem dubious. Even if the Phoenix's abilities still worked after its own demise, Roshi's willingness to hand it over so easily in the first place suggests he has some other means of staying alive.
Is it the Dragon Balls?
An easy solution would be to say that Roshi used the Dragon Balls to wish for immortality. This wouldn't be much of a stretch, as several characters such as Vegeta, Frieza and King Piccolo sought the Dragon Balls for the same reason, with some even succeeding. However, this seems unlikely, as Roshi was unaware that the orbs had any value before meeting Goku and Bulma. While he initially kept the three-star Dragon Ball as an accessory, Roshi gives it to Bulma in exchange for a peep show. While he does eventually find out about their wish-granting capabilities, he's never shown using them for anything personal.
So Is He Just... Really Old?
Unthinkable though it may be, is it possible that Master Roshi can just naturally live for centuries? After all, he's the younger brother of Fortuneteller Baba, who herself is over 500 years old. While this might suggest that the Roshi family tree simply has absurd genetics, Baba likely gets her long life from some sort of magic due to her powers over the dead and Other World travels. Likewise, another comment from Roshi's chelonian compatriot Turtle implies that there's some external factor keeping the Hermit around.
The Fountain of Youth
When Goku and Chichi go to Master Roshi for assistance finding the Bansho Fan, he offers help in exchange for further lewd favors from Bulma. He attempts to coax pity for his lecherous ways by calling it a final wish in his last years. However, Turtle calls him out on this claim, stating that Roshi drank from the Fountain of Youth. While this seems like a straightforward answer, the Japanese version says that he drank "an eternal youth potion." If one takes the latter as the more definitive version, it begs the question of what the potion's ingredients are.
In Episode 75 of the Dragon Ball Super anime, Roshi sends Goku and Krillin on a training exercise to retrieve a mysterious herb. When they return in Episode 76 with a barrelful of this "Paradise Herb," Roshi guzzles them down, saying that he can "feel his vitality returning" and that he "could live for another 1,000 years." Although this could be purely idiomatic, the gusto with which he consumes them suggests that he's being literal.
Still, while this episode seems to finally solve the mystery, it's possible that the Fountain of Youth, Immortal Phoenix and Paradise Herbs are all just part of his death-defying treasure hoard. It would certainly be in character for him to have such a stockpile hidden away. If so, Roshi may just stay alive until the end of the world, giving out training advice and harassing young women forever.