Scripture does not explicitly say exactly how tall the Tower of Babel was. According to scripture, the people building the city of Babel wanted to built a tower that reached the heavens.
The story about the Tower of Babel begins in verse 11:11 of Genesis, the first book of both the Hebrew and Christian bibles. God sees the people in the process of building the tower and realizes that with a common language, there was nothing that humankind couldn't accomplish.
To keep them from attempting to be God-like, God scattered the people building Babel across the Earth. They stopped building the city, and languages became confused.