＊だって：だ＋という：The dictionary translation is just “but”.
I would like to give my idea which would translate as: “Talking about this/ mentioning this”, which will be followed up by a sentence that tries to rebut the hearsay his colleague dropped before.
＊30年間（さんじゅうねんかん）： Thirty years (timespan)
＊が：Particle the “Subject” to the Verb ない that comes after.
＊ない：Negative form of だ/です.
Example: なになにがない。-> Something doesn’t exit.
＊なん：Particle implying ??? (please help me here)
＊だろ：Informal form of でしょう：“it seems like this/ I think it is like this”.
PUTTING IT TOGETHER:
30年間手術：“thirty years of surgery”
ミスがない人：“a Person who doesn’t make mistakes”
なんだろう……？： “isn’t it like this?”
“But having 30 years of surgical experience means they should not make any mistakes anymore, right?”
“But people who have 30 years of surgical experience don’t make any mistakes anymore, do they?”
Looking forward to suggestions and corrections!
Correction by @Naphthalene
“But he is a person who hasn’t made a surgical mistake in 30 years”