but that Israel who pursued a law that would lead to righteousness did not succeed in reaching that law. (ESV)
It would be hard to find a more "religious" people than the people of Israel. To this day there are Jews who take their Sabbath day very seriously. When it comes to feasts and festivals the Jews were passionate about celebrating them all. They looked to the Law of Moses as their guide. The sad reality is that they pursued a law that they could never attain.
If you want to know what the Jews were like whom Paul was talking to in the book of Romans it would be helpful to look at the Pharisees of the Apostle Paul's day. Jesus had many encounters with them. The Pharisees were the spiritual leaders of the people of Israel. If anyone would know how to pursue righteousness it would be the Pharisees. Listen in on some words Jesus had for them:
 “The scribes and the Pharisees sit on Moses' seat,  so do and observe whatever they tell you, but not the works they do. For they preach, but do not practice.  They tie up heavy burdens, hard to bear, and lay them on people's shoulders, but they themselves are not willing to move them with their finger. (ESV)
Why were the burdens so hard to bear that the Pharisees were giving to the people? Because they were works done to try and somehow attain righteousness. Jesus Christ came to set people free from the bondage to works. Jesus himself fulfilled the entire law of God in his life and died for the sins of all those who broke the Law on the cross. Works do not attain righteousness. Christ's work does.
Israel worked for what they could never achieve. All who recognized their own futility and placed their faith in the one who worked for them were saved. Are you still trying to work for your righteousness. Stop. Place your faith in the one who works for you. His work on the cross was enough.