Talking about startups, does this just mean keeping your eyes on your numbers and doing frequent review or are we talking about something on a more personal level?
I think by definition, self-awareness means something on a more personal level. You have to understand not only the market you're in, the business you're trying to start, your product and your customers, but also must be keenly attuned to your own methods, your "chemistry" with your partners, and the _way_ you approach the business, not just the approach itself.
At least that's been my experience as a self-proclaimed "self-aware entrepreneur" ;)