Where did Abraham the Jew live?

Answer 1

Abraham was born in Ur (Iraq; Babylonia) and lived most of his life in Israel (Canaan).

Other information:

Abraham, tenth-generation descendant of Noah, of Hebrew lineage, was the son of Terah, uncle of Lot, father of Isaac, grandfather of Jacob, and forefather of the Israelites. His story is in Genesis ch.11 (end), through ch.25. Jewish tradition states that he was the first to actively spread belief in One God; and it is in his merit that Jews continue to exist (Genesis 18:19, and ch.17).

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Abraham came from ancestry that had been aware of God a couple of centuries earlier but had afterwards slipped into idolatry (Joshua 24:2). By the time of Abraham, the area where he lived was full of pagan cults; they were polytheistic, worshiping multiple deities.
Abraham became the first to advance the idea of ethical monotheism: the worship of One God, and the appropriate ethical code of conduct.

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Nimrod, the idolatrous tyrant, had brought Abraham's father (Terah) from the Semitic ancestral seat near the confluence of the Balikh and the Euphrates, and instated him in a position of power in his army in the royal Babylonian city of Ur, where Abraham was born. Nimrod persecuted any who would question his idolatrous cult.
The Kuzari (Rabbi Judah HaLevi, 1075-1141) states that Abraham was gifted with high intelligence; and, as Maimonides (1135-1204) describes, Abraham didn't blindly accept the ubiquitous idolatry. The whole populace had been duped, but the young Abraham contemplated the matter relentlessly, finally arriving at the conclusion that there is One God and that this should be taught to others as well. This is what is meant by his "calling out in the name of the Lord" (Genesis ch.12). As a young man, he remonstrated with passersby in public, demonstrating to them the falsehood of their idols; and our tradition tells how he was threatened and endangered by Nimrod.
Subsequently, Terah relocated to Harran; and it is here that Abraham began to develop a circle of disciples (Rashi commentary, on Genesis 12:5).
Later, God told Abraham in prophecy to move to the Holy Land, which is where he raised his family.
He continued his contemplations, eventually arriving at the attitudes and forms of behavior which God later incorporated into the Torah given to Moses.
Abraham became the greatest thinker of all time. His originality, perseverance, strength of conviction, and influence, cannot be overestimated.
Abraham, with God's help, trounced the supremacy of the evil Nimrod.
He received God's promise of inheriting the Holy Land (Genesis ch.13).
He strove to raise a family (Genesis ch.15, 17, and 24) which would serve God (Genesis 18:19); and God eventually blessed his efforts, granting him numerous descendants (ibid., ch.16, 21 and 25), in keeping with His promise (Genesis ch.17).
Abraham founded the Jewish people and lived to see his work live on in the persons of Isaac and Jacob; and he taught many other disciples as well (Talmud, Yoma 28b).
He saved the population of the south of Canaan from invading foreign kings (Genesis 14); and he was feared by neighboring kings (ibid., ch.12 and 20).
Abraham gave tithes (Genesis ch.14), entered into a covenant with God (Genesis ch.15 and 17), welcomed guests into his home (Genesis ch.18) unlike the inhospitable Sodomites (Genesis ch.19), prayed for people (Genesis ch.18), rebuked others when necessary (Genesis ch.20), eulogized and buried the deceased (Genesis ch.23), and fulfilled God's will unquestioningly (Genesis ch.22). He became renowned as a prince of God (Genesis 23:6).
The gravesite of Abraham, Isaac and Jacob and their wives (Genesis 49:29-32) is located in Hebron and has been known and attested to for many centuries.
All of the above practices of Abraham were based upon the ways of God, which Abraham understood through his contemplations. These, and similar personality traits, were the teachings of Abraham and his descendants (unlike idolatry, which had no moral character; with worship of the gods accompanied by things such as human sacrifice, "sacred" prostitution, and animal worship).
It is therefore clear why God expresses His love for Abraham (Isaiah 41:8) and calls Himself the God of Abraham (Genesis 26:24), and says that Abraham obeyed Him fully (Genesis 26:5). And this is why Abraham is credited with having begun the religion which became known as Judaism. (However, Abraham and his descendants observed their traditions voluntarily, until the Giving of the Torah to Moses 3325 years ago, when God made it obligatory.)

Answer 2

Tradition holds that Abraham lived in the land of Israel.