According to Hanyu Wailaici Cidian (Dictionary of Chinese Loanwords), bóshì (博士) with the modern sense "doctor" is derived from the Japanese hakushi, which in turn is of "Ancient Chinese" (古代汉语) origin. I'm aware that the Japanese have coined many terms to translate Western concepts, many of which are actually ancient Chinese terms that are assigned new meanings. But I'm puzzled with the concept of "Ancient Chinese". What kind of Chinese was that? Was it Old Chinese, or Middle Chinese as Wikipedia suggests? I doubt that a disyllabic word like 博士 was coined during the times of these languages, because I've read that they were highly monosyllabic. Also, Middle Chinese was only spoken around the 10th century and earlier, so what are later stages of Chinese called? Is it possible that the Japanese employed words like 博士 or 文化 from those stages?
古代汉语 is everything that is not 现代汉语. More precisely, we have 上古汉语 (Old Chinese: Oracle bones, Warring states, Qin; Classical Chinese: Spring and autumn, Han）, 中古汉语 (Middle Chinese: Tang, Song) and 近代汉语 (mostly late Qing).
博士 was coined in the Warring states period, according to zdic.net. Disyllabic words have always existed, although the ideal used to be monosyllabic word characters.