"For the law of the Spirit of life in Christ Jesus hath made me free from the law of sin and death. For what the law could not do, in that it was weak through the flesh, God sending his own Son in the likeness of sinful flesh, and for sin, condemned sin in the flesh: That the righteousness of the law might be fulfilled in us, who walk not after the flesh, but after the Spirit. (Romans 8:2-4)"
John Calvin is right in saying that, "The word flesh is to be taken still in the same sense, as meaning ourselves. The corruption then of our nature renders the law of God in this respect useless to us; for while it shows the way of life, it does not bring us back who are running headlong into death." Calvin's Commentaries, Vol. 38: Romans, tr. by John King, [1847-50]
It takes a perfect person to obey the perfect law of God and the only person who has ever done so is Jesus Christ. It is not that the law of God cannot justify someone, it is that there is no one except Jesus Christ that is able to obey it perfectly to be able to be justified by the law of God.
The righteousness of Christ is what made his sacrifice on the cross perfect. Though he was sinless he took upon himself the infirmities of our sinful flesh, including a painful death. Not only did Christ obey what we cannot obey, he defeated what we cannot defeat, namely death. When someone repents of their sins and receives him as their Lord and Savior they are reaping where they have not sown; they become beneficiaries of the rewards that come from his obedience, sacrifice, and triumph over the grave.
It is only through Christ that the righteousness of the law can be fulfilled in us. Left to ourselves, we will fall short of being justified by the law of God since we cannot keep it perfectly through the infirmities of our flesh. Christ is our only hope of ever being justified before God and being able to stand before him as if we never sinned.
When you are presented with the law of God, it will expose your inability to justify yourself and reveal your need for Jesus Christ. Even what we may consider petty sin (lies, stealing, etc) separates us from God and we all need Christ for reconciliation, justification, and eternal life.