Learn Turkish: Vatanı ölmek pahasına savundular. - They defended the homeland at the cost of dying.

The word pahasına acts as a postposition meaning "at the cost of" or "at the risk of." When the "cost" is an action, you simply use the standard infinitive form of the verb ending in -mAk. So, ölmek (to die) + pahasına gives us "at the cost of dying."

What is the literal grammatical breakdown of the word pahasına?

It is built from the noun paha (price, cost, or value). We add the third-person possessive suffix -sı (its cost), the buffer consonant -n-, and the dative case ending -a (to/at). So it literally means "at its cost."

Could I use a possessive verbal noun here, like ölmeleri pahasına (at the cost of their dying)?

You grammatically can, but it is not necessary and sounds a bit clunky here. When the person taking the risk is the same as the subject of the sentence (they are the ones defending, and they are the ones risking death), the plain infinitive ölmek is the most natural choice. You would typically use a possessive form like ölmem pahasına (at the cost of my dying) if the subjects were different, for example: "They defended it at the risk of my death."

Why is vatan (homeland) in the accusative case as vatanı?