The only persont that can answer that one is Just Asking. I wouldn't know. It works for me. There is not that many people on it, but there seems to be enough of us, even with those coming and going, to keep it alive.
By whether or not they meet their revenue goal. That goal will vary from site to site. It may mean just enough to break even or it may be a substantial amount.
The revenue goal is derived from advertising income. I believe advertising income is derived from the number of people who join and the frequency with which they ask and answer questions.
It depends on the site. I love reading about the motivations behind site creators. There are lots of interviews out there with the creators of Answerbag and Experience Project. It's my understanding that AB's primary goal was to be a huge searchable database of information. From that respect, I bet its creator thought it was a success when AB was at its prime. When Demand Media took it, they undoubtedly cared primarily for financial gain. It was a flop, so they canned it. With EP, the owner created it to be a support site. I'm sure he probably felt it was successful too. It also got sold to an entity that was concerned with revenue. Goodbye, EP. Mind you, I have no insider information on either of those sites. I only know what has been publicly published and base my hypothesis on that.
It's my belief that each site has a heart and a purpose. When the site is in the hands of a non-corporate entity (typically the original creator), success is generally measured in whether the site is fulfilling its purpose. When it goes corporate, it usually becomes about profit.
If you want to know how I measure success, it's in the level of respect and tolerance I see. My only goal is to have a venue where people are comfortable and free to talk without fear of personal attacks. I don't care if there are five million people here or five. I don't care if the site turns a profit. If ideas are being shared and people are decent to each other, that's success to me.