If one claimed that a particular scholar was "above average" or
"noted" in their field, is there any good metric by which to support
or deny such a claim?
The only "generally approved" quantitative metric is the h-index. H-index is a metric, is OKfor your task as it allows you to define above or below average. As a matter of fact, this is the way some national educations systems stamp their professors as good enough for tenure. It is also agreed that it is not "good enough" - famously, Peter Higgs, 2013 Nobel in Physics, would fail miserably a ranking based on h-index only, as he published very few paper, although with huge citation count. Also, h-index is a measure of lifetime achievements, thus needs to be corrected for the academic age. Which brings us to the next point.
Is there a generally accepted way to indicate that a particular
professor or scholar is outstanding, or above average, in their field?
I understand there are certain indicators, such as chairs, endowments,
prizes, etc. But these don't really seem to help to compare one
scholar to another, except in a sort of gross, simple count way (i.e.
one professor has had more chairs than another)
Other, mostly qualitative metrics are regularly used, consciously or not, in academic's minds, although no official ranking exist. I will mention a few, the ordering only reflecting the stage in an academic career:
1. Institution where PhD has been obtained
2. PhD supervisor
3. national prizes
4. national grants
5. number of PhD students supervised
6. chairs at institutions or conferences
6. international prizes
7. academic success of PhD students mentored
8. more I could not think about now :)
Is it theoretically possible to create a "ranking" of professors in
their fields, by some metric?
Of course it is, there is entire field about it called Scientometrics. You have to 1) fix for h-index known limitations 2) combine with the variables above to come up with a more comprehensive algorithm that will rank any researcher in any field. The reasons why this has not been done before are twofold. First, it is not easy at all to define objectively how much every metric listed here should weight in the ranking algorithm. Second, and most importantly, academics rank every day for jobs, promotions, accepting papers or conference contributions, prizes etc. However, they prefer their ranking algorithm to suit their individual minds, rather than adopting a common framework.
Could their h-index serve as such a metric?
As described above, h-index has many limitations that make it impractical for most purposes. But an entire field of research exists around it - Scientometrics - so rest assured there will be developments.