Let's see the context:
Amos 9:11“In that day I will raise up
the booth of David that is fallen
and repair its breaches,
and raise up its ruins
and rebuild it as in the days of old,
12that they may possess the remnant of Edom
and all the nations who are called by my name,”
declares the LORD who does this.
they may possess
Verb - Qal - Imperfect - third person masculine plural
Strong's Hebrew 3423: To occupy, to seize, to rob, to inherit, to expel, to impoverish, to ruin
Noun - feminine singular construct
Noun - proper - masculine singular
The verb is in Qal Imperfect followed by an accusative case, ie, the direct object of a transitive verb.
2 inherit, followed by accusative of person = be one's heir Genesis 15:3,4 (twice in verse) (JE); שִׁפְחָה כִּיתִֿירַשׁ גְּבִרְתָּהּ Proverbs 30:23 (or, dispossess?); absolute be heir, followed by עִם person = jointly with Proverbs 21:10 (E); (הַ)יּוֺרֵשׁ (the) heir 2 Samuel 14:7; Jeremiah 49:1 ("" בָּנִים); Micah 1:15 (= possessor, captor), where paronom. with proper name, of a location מָרֵשָׁה; inherit persons, as slaves לָרֶשֶׁת אֲחֻזָּה Leviticus 25:46 (H).
Specifically, what does "possess" mean here, and what does it mean to "possess" a certain group of people?
It means that Israel would inherit the land of Edom, its closest relatives, plus all the gentile nations who believe in God. It is a prophecy of the Messianic Kingdom, the Kingdom of God.
But now that you have been set free from sin and have become slaves of God, the benefit you reap leads to holiness, and the result is eternal life.
So that we may work the work of God.