Learn Turkish: Saflık bakımından yakut nadir bir taş. - In terms of purity, the ruby is a rare stone.

Bakımından literally translates to "from its aspect" or "from its point of view." It is built from bakım (aspect/viewpoint) + -ı (possessive marker linking it to the bare noun saflık) + -n- (buffer consonant) + -dan (from). The ablative -dan suffix is what creates the "from the perspective of" meaning.

Could I use açısından instead of bakımından in this sentence?

Yes, saflık açısından and saflık bakımından are perfectly interchangeable here. Both mean "in terms of purity" or "from the perspective of purity."

Just to refresh, why is the word order nadir bir taş instead of bir nadir taş?

In Turkish, when a noun is modified by both a descriptive adjective and the indefinite article bir, the adjective almost always comes first: [Adjective] + bir + [Noun]. Saying bir nadir taş sounds unnatural.

There is no verb here. Why isn't there a "to be" suffix like -tır at the end of taş?

In everyday conversational or standard written Turkish, the third-person copula (the "is/are" suffix) is very frequently omitted. You could absolutely say nadir bir taştır, but adding -tır makes the sentence sound more formal, like an objective fact you would read in an encyclopedia.