"Precarious" and "normal" are not two words that you would usually put together, but the juxtaposition of two seemingly contradictory things can be read as funny or clever or both, depending on the context. "Precariously normal" suggests that (as you guessed) his life is continually on the verge of becoming abnormal.
This kind of contradiction also creates a sort of dramatic tension, so we can assume the writer is about to describe his life in some detail. It would be awkward for the writer to just drop something like this in a story without adding the supporting context.
To give another example. Suppose, say, there is a guy who builds a wall, and he goes on and on about how it's a really great wall, the best wall, how it would stop anyone from crossing it, and how it would be the ultimate protection. It would be, he said, like a "solid steel barricade".
And then videos appear showing hundreds of people easily crossing over the wall with ladders, or cutting through it, or digging under it. In response someone writes:
"Solid steel"? Looks more like a tissue paper barricade to me.
By itself a "tissue paper barricade" obviously means the barricade is not effective at what it's supposed to do, but it's not funny or clever (or possibly just stupid and insipid) without the context to explain the contrast.