The correct answer is 'has been doing'.
The sentence requires the use of the present perfect continuous tense because it describes an action (what Mark has been doing) that started in the past (since he quit his job) and is still continuing (or has relevance to the present). The present perfect continuous tense is formed using 'has/have been + verb-ing'. Because 'Mark' is a singular subject, we use 'has'. Therefore the correct answer is: 'What has Mark been doing since he quit his job?'