I wasn't thinking and put "Adam is seventy years old" for the translation and it still marked it correct :D
I'm not sure, but it could be that yetmiş and yaş are combined in a compound noun, which requires the genitive case suffix -ı on yaş. To imply that someone is of (at) that age would then require the locative case suffix -da, for which the genitive case suffix requires the buffer consonant -n, hence yaşında. But here I'm just combining what I've learned so far, so it could be wrong.
Sara, i have the same question and looked for an answer in different sites of turkish grammar. I have found that in https://turkishexplained.com/nouns.htm. If i understand well, "yetmis yasinda" are indefinite compound nouns= "his 70th age" "yas"+ "i" possessive siffix + "n" buffer before "da" locative suffix. Why "yetmis" without genitive suffix? They give this example: "araba kapısı"= "car door" as indefinite compound nouns and "arabanın kapıları"= "the car's door" as definite compound nouns. . What do you think about that?