What were Abraham's circumstances when God promised him a son?

Abraham was a hundred years old, and his wife Sarah was ninety. God kept telling him he would have a son and that a nation would come from his body. This seemed against hope and common sense because of their age and the physical impossibility of reproduction.

Les teaches that Abraham, at about one hundred years old with a wife of ninety, was promised by God that he would have a son and that a nation of people would come from his body. This was against hope, meaning against common sense and the physical understanding of reproduction. Yet Abraham believed in hope, knowing that no matter how ridiculous it seemed to the human intellect, God was going to bring it to pass. The promise that he would become the father of many nations was according to what was spoken, 'so shall thy seed be.' Les also notes that out of Abraham, Sarah, Jacob, Isaac, and Ishmael came all the nations of people on the earth.

Romans 4:18

Who against hope believed in hope, that he might become the father of many nations, according to that which was spoken, So shall thy seed be.

Why did Abraham keep believing God would give him a son when it was physically impossible?

Abraham kept believing because he trusted God's repeated promise, not physical possibility. Though he was a hundred and Sarah ninety, he had genuine hope, knowing that God, who calls things that are not as though they were and quickens the dead, would somehow bring the promised son to pass.