To exact something (such as payment or revenge) is to demand it and get it, especially by using force or threats. Exact is also used in phrases like "exact a terrible toll" and "exact a high/heavy price" to say that something has caused a lot of suffering, loss, etc.

// The novel's protagonist is undeterred by either time or circumstance in her effort to exact revenge.

// Our mistakes exacted a heavy price.

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"... Milo did fake his death last season and has ever since been lurking in the shadows, waiting to exact revenge on Iris and others." — Noel Murray, Vulture, 28 July 2024

To exact something is to not only demand it, but also obtain it. The most common things exacted—revenge, retribution, and that ilk—often require physical force, but other things exacted—such as penalties and prices, promises and concessions—can be obtained with gentler forms of persuasion. The variation is present in the word’s Latin ancestor too: exigere means “to drive out; to demand; and to measure.” Do not confuse the verb exact with the more common verb extract. Extract is primarily about removing something, and need not involve a demand: a dentist extracts a tooth from (we hope) a willing patient, and extracting juice from an orange carries no connotation of insistence. Note, though, that there are cases in which either verb can be used: confessions, for example, are sometimes said to be exacted or extracted, with both typically implying significant effort by the one obtaining the confession.