The scriptures state that those who walk in justice will indeed see the face of God and live. There is also a famous case that provides the exception to the rule that one cannot see the face of God and live.
This dramatic event is part of the vision of Isaiah 6 in which the prophet "saw the Lord" and feared that he was doomed because he was a man of "unclean lips." Granted that the word "face" is not mentioned, but that is clearly the implication. Thankfully he was given special grace and went on to a successful career. In addition:
Psalm 11 provides perhaps a clear example of how a person can see God and live:
The Lord is in his holy temple;
the Lord’s throne is in heaven. God’s eyes keep careful watch;
they test the children of Adam... 7 The Lord is just and loves just deeds;
the upright will see his face.
Psalm 15 carries a similar message:
Lord, who may abide in your tent?
Who may dwell on your holy mountain? 2 Whoever walks without blame,
doing what is right,
speaking truth from the heart... Whoever acts like this
shall never be shaken.
In Psalm 18 the author speaks of God rewarding him because of "my righteousness and the cleanness of my hands." In his temple, the LORD heard the prayer of the distressed psalmist and granted him a vision in which he describes God's face:
Smoke rose from his nostrils,
a devouring fire from his mouth;
it kindled coals into flame (vs. 9)
There is no sense in these psalms that achieving a state "clean hands" and moral uprightness is impossible. Those who act justly will see God's face. In the case of Psalm 18, this has actually happened, and in Isaiah's case he saw the Lord and yet lived because he was given grace.
Conclusion: Christians usually deny that a state of uprightness is possible outside of Christ. Paul argues persuasively that everyone sins and needs to be reborn in Christ. Jesus himself declared that no one comes to the Father by Christ. However, it cannot be denied that the scriptures also speak of another way. Isaiah saw God's face and lived, as did the author of Psalm 18. Psalms 11 and 15 speak of waking uprightly as the normal state for pious Israelites. Thus, Jesus' answer is is not the only one given in scripture.