what are gentlemen-scholars？

Saying someone is "a gentleman and a scholar" basically means "nice (polite) and smart".

Need the context!

The phrase sounds as though it refers to the British Education system prior to WW1 when universities (predominantly Oxford and Cambridge) and Public Schools (such as Eton, and Harrow) were populated almost exclusively by "gentlemen" (from the rich upper classes). The word "scholar" was used to refer either to any schoolboy or specifically to one who had obtained a "scholarship" to Oxford or Cambridge (providing him with remission of fees and allowing him to use the title "scholar" on his name-plate at the college).

Good answers.

A "gentleman" defined in the terms fresco uses, and of those times, could also refer to someone of independent means, who does not have to work for a living and who studies his subject(s) purely for the love of it. Otherwise stated, an amateur, in the original meaning of the word.

McTag's point is well taken. Those who did not have to labor for their sustenance, well before the rise of the public schools and the opening of Oxford and Cambridge to those who were not intended ot be members of the clergy were said to be "gentle born." A gentleman or gentlewoman was a member of that class. A graduate of either Oxford of Cambridge for the early centuries was intended for the clergy. Although this might include the gentle born, by and large the graduates were looking for benefices which would support them, they were looking for appointments to what were literally described as livings. Someone who is gentle born who determines to also be a scholar is, as McTag points out, someone who does it for the love of thing, and not to get their daily bread.