Roshi and Master Shen (Crane) were rivals while growing up. But why did Crane not die of old age?
A major thing to keep in mind is that Roshi is an unreliable narrator (and his turtle isn't terribly smart by human standards). He routinely lies and twists half-truths to help people get through something tough, motivate his students to work harder, or get something he wants that he thinks people otherwise might not give him. It's a generally benign sort of thing, serving his mostly comedic perversions or serving some sort of "greater good".
As Gary mentions, in the original series there is mention of an Elixir of immortality/Fountain of Youth, and Roshi at one point tries to console Tien by pointing out that he (Roshi) is immortal, so there's nothing to really worry about. This was another lie or half-truth, though, as Tien was soon knocked out and Roshi was killed by the strain of the technique he used to try to save the day. So, as indicated, he twisted the truth, or outright lied, to help someone out. Most people at this point presumed that in fact the elixir/fountain only stopped him from dying of old age (and possibly other natural causes), but suicide/exhaustion/getting eaten/laser beams to the face/etc. would still be lethal in the usual fashion.
In Super, Roshi sends Goku and Krillin out to fetch some paradise herb in exchange for teaching them a super secret technique. The technique was a half-truth, though. It was meant to motivate them to do the quest, and thereby help Goku increase his control of his energy and for Krillin to overcome his (very justified) fears. So they were taught something useful for combat, but not really a "technique" (and certainly not one of Roshi's martial arts techniques). He had also stated that eating the herb would increase his lifespan by another thousand years. Whether that was another lie, or a bit of dramatic license, or not...we don't know. But it stands as the current best explanation for Roshi's long life. Presumably, Master Shen and his brother also partook of this herb.
Alternative options do exist, however. Demon King Piccolo used the dragon balls to acquire Eternal Youth, which was inherited by Piccolo. This prevented Piccolo from aging once he hit his physical prime (a mere 4 years later), but neither of them had true immortality and could be killed by means other than aging. Conceivably Roshi, Shen, and others may also have made such a wish at some point.
A standard martial arts trope is the long-lived master, where their supreme skill and spiritual development gives them an unnaturally long lifespan; this may even be a specific "technique/skill" of theirs. Such a technique may have been found by Roshi/Shen/General Tao/Baba and used amongst them, only they found it later in life and it does not revert your body to an age before you acquired it. Part of Roshi's character is how he at once serves this role of a transcendent master of martial arts while also being a perverted, lazy, goofball.
And there's the fact that if he is in fact several hundred years old, then he may have gone through a lot of stuff, and the reason he has lived so long is buried somewhere in all of those adventures, possibly stretched across several of them in a complex fashion, and he may not even remember them all. After all, he once had true immortality by owning a Phoenix (which ironically died, and the immortality was lost). Which is to say there's plenty of story telling room to "retcon" the reason for his long life and insert some new justification(s) later. All of which serves the handy purpose of making others distrusting of his claims, and difficult for others to achieve what he has (which could be dangerous to the world, as well as those attempting the feat).
This is a mistake which was overlooked and never addressed in the series. It was initially assumed that they drank the Elixer of immortality from the Fountain of Youth. However, Roshi later claims this doesn't really exist. So yes! There is no logical reason explaining why the Crane Hermit is as old as he is.