Someone who says something like the first sentence generally means something like "questions that are significant and in an area I am concerned with"

If you don't have any open questions in that category, then you aren't really living as an intellectual.

In science questions are like a hydra. After solving a scientific problem you often have more questions than you had when you started.

Schwartz's article on the issue is quite illustrative. If you can't deal with the emotional effects that come with looking at an open question and having it open for months and years you can't do science.

You won't contribute anything to the scientific world of ideas if you can only manage to concerned with an open question for an hour and not for months and years. Of course there are plenty person in the real world who don't face questions with curiosity but who in pain when dealing with them. To me that seems like a dull life to live. because the question doesn't concern themselves with living an intellectual life.