Perrenial Professions?

An immortal living through history needs to move around to conceal his longevity and also needs to earn a living. Some fictional immortals practice different lines of business down the centuries. At least two that I know of identify a kind of work which they think will always be needed and stay with that.

For Poul Anderson's example, see More On Patulcius.

Neil Gaiman's Bernie Capax:

""From time to time he's done other things, but mostly he's been a lawyer of one kind or another.
"People always need lawyers."
-Neil Gaiman, The Sandman: Brief Lives (New York, 1994), Chapter 3, p. 2, panel 1.

So what happens to them? Capax dies when a wall falls on him and Patulcius outlives the need for archivists.