Why did Israel not enter the promised land, and what kept God from destroying them?

Israel failed to enter the promised land because of unbelief at Kadesh Barnea, turning back into the wilderness instead of going in. God did not destroy them as He did Sodom because a remnant of true believers always existed, and that remnant caused God to spare the nation.

Les explains that Israel was constantly turning back in unbelief, beginning especially at Kadesh Barnea, where they could have gone in and had the land of promise. Instead, in unbelief they turned around and went back into the wilderness. Yet there has always been a remnant of true believers. Had it not been for that remnant every time, God would have had to destroy the nation of Israel like He did Sodom. Likewise, at the end of the age, a remnant of Israel will cause God to spare the nation.

Romans 9:29

And as Esaias said before, Except the Lord of Sabaoth had left us a seed, we had been as Sodoma, and been made like unto Gomorrha.

What happened at Kadesh Barnea when Israel could have entered the promised land?

At Kadesh Barnea, when Israel could have entered and possessed the land of promise, the people turned back in unbelief instead. They reversed course and returned to the wilderness. Their unbelief kept them from going in at that time, though a remnant of true believers remained.