What does Romans 3:10 mean?

Paul has proclaimed in the previous verse that everyone, Jew and non-Jew alike, is "under sin." He is showing that Jews cannot hope to be shielded from God's judgment for their personal sin, simply because they belong to the nation of Israel. Every person, Jew or Gentile, will be judged by God on the basis of their own right and wrong choices. By that standard, every person, Jew or Gentile, will be found to be "under sin" or guilty and deserving of God's anger.

Now Paul begins to back up that claim with a series of quotes from the Hebrew Scriptures—what we now call the Old Testament. He wants to show that this idea of universal sinfulness is not a new idea. David wrote the same thing in Psalm 14:3, "There is none who does good, not even one." The following two verses will complete the reference, which thoroughly eliminates the possibility that any person has ever "done good" sufficient to make them holy in the eyes of God.

This verse is often-quoted, and for good reason. The idea that all people—without exception—are in need of salvation is a key point of the gospel. Scripture leaves no room for anyone to claim they are "good enough" to deserve heaven. Paul will return to this same idea, using different phrasing, in Romans 3:23.