I think he means well but he's not the brightest bulb on the Christmas tree.
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Matthew 8:11–12
11 And I say to you that many shall come from the east and the west,(Gentiles) and shall sit down with Abraham, and Isaac and Jacob in the kingdom of heaven:
12 But the children of the kingdom shall be cast out into the exterior darkness: there shall be weeping and gnashing of teeth. biblia.com/bible/douayrheims/Mt8.11-12

John 14:6 "Jesus saith to him: I am the way, and the truth, and the life. No man cometh to the Father, but by me."

Pretty straightforward. Barron is not Catholic. He's a modernist, de facto protestant with holy orders.

Comment from Fr. Brian W. Harrison, MA, STD

In answer to Ben Shapiro’s appeal to his own good works – upholding the 10 Commandments and 603 other Jewish laws – on which he bases his expectation of salvation, we needed to hear from Bishop Barron the clear New Testament teaching of St. Paul that good works without faith in Christ are not sufficient for salvation. E.g., “We who are Jews by nature and not sinners among the Gentiles, . . . know that a person is not justified by works of the law, but through faith in Jesus Christ. We have believed . . . that we may be justified by faith in Christ and not by the works of the law, because by works of the law no one may be justified” (Gal. 2:16). Many other passages from St. Paul and the Gospels could be cited to the same effect.
Also, Vatican II doesn’t in fact say that atheists can be saved if they are still atheists (i.e., faithless) right up to the moment of death. (The relevant passage in Lumen Gentium is one of the Council’s well-known ambiguities that have led to much confusion in the last half century.) Scripture flatly contradicts the idea that a faithless person can be in the state of grace and thus ready for salvation, for “Without faith it is impossible to please God” (Heb. 11:6). Indeed, Vatican I condemned as heresy the idea that even the natural knowledge of God accessible through human reason is equivalent to the divine (supernatural gift) of faith that is necessary for salvation, since (the Council affirms) it is “necessary to divine faith that revealed truth be believed because of the authority of God who reveals it” (Canon 2 on “Faith”, Denzinger 1811). Much less, then, can a person be saved who at the moment of death lacks not only supernatural faith but even a natural, rational, belief in God.
If I had been the one talking to Ben Shapiro, I would have honestly answered negatively his question as to whether he could be saved as a Jew, referring to the above teaching of St. Paul and the infallible Catholic Magisterium about the necessity of faith in Christ for salvation. But I would also add what St. Thomas Aquinas, in his wisdom, taught about this: if a non-Christian perseveres in seeking truth and striving to follow the natural moral law in the light of his conscience, then somehow or other, before the moment of death, God will give him the grace of enlightenment – maybe via an unexpected missionary, an angel, or an interior revelation – so he can repent of sin, explicitly believe at least the most basic truths of the Christian Gospel, and thus reach eternal life.
A multiplicity of recorded and clinically examined “Near-Death Experiences” over the last half-century have shown indisputably that people can sometimes have life-changing conscious experiences when close to death, even while they appear totally unconscious to bystanders at the hospital bed, and even to instruments recording their brain activity. So the fact that a morally upright Jew or other non-Christian does not show any indication of conversion to Christ on their deathbed is by no means proof that such a conversion has not taken place.