The quote about the market being irrational longer than you can remain solvent is very relevant for investors using leverage and/or short selling. As an example, in the late 90s it was very obvious that many dot-com stocks were overvalued, and that eventually the bubble would burst. But shorting a dotcom business was incredibly dangerous, because it could double or triple or go up 10X for no good reason, and your short position that was 3% of your portfolio becomes 30% of your portfolio. Even if you ended up being right, you could end up getting margin called and forced to liquidate your position. Similarly, anyone investing with leverage can get in trouble easily if stocks go down 50%. The perfect example of all of this is the hedge fund Long Term Capital Management in the late 90s, which blew up because, although they were correct in their arbitrage bets, they were highly levered, and the market moved against them in the short term and they got screwed.
As an individual investor not using leverage and investing for the long-term, the quote really doesn't apply. The "long-term" part is obviously key, however, because we've had periods where stocks have been flat or negative for a decade or two (i.e. ~1929-1950, or 2000-2010, etc).