Takemi's probably 27/28. Last I heard, which was admittedly right after I graduated high school, Japanese doctors needed 6 years of schooling, 2 of residency- with various vicious exams and licensing requirements in the way.

By virtue of being an attractive female anime character with an "alternative" style and/or attitude, Takemi is guaranteed to have been a dark horse that tore through the exams and qualifications first try. Then, we know she had some experiences running a clinic prior to MC meeting her (though they could have been on residency).

But the biggest piece of evidence for her being under thirty is that you can date her in the first place. The "women need to be married by 30" thing in Japan seems really deep set.