Failure is a relative question. I've been part of 2 startups that still exist as profitable ventures, but in my opinion failed.
#1 - Market research company founded in 1989 (I joined in 1990) and it still exists today. I left in 1994 when it was 40 employees. It is presently at 18 employees. It still pays the owners bills but toils in obscurity with no growth.
#2 - Franchise services company founded in 2004 (I joined on at the founding) and it still exists today as well. I left in 2009 and it was 3 employees. It is still 3 employees and much like company #1 it has virtually no growth.
While both companies are profitable, barely, they both suffer from what I call "zombie mode syndrome." They make enough money to stay afloat but without any organic growth they can't fund expansion nor attract investors. And for that, I believe they are failures.