Does the Bible teach that OT Israelites were saved by obeying the Law?

The Work of Christ

[21] But now the righteousness of God has been manifested apart from the law, although the Law and the Prophets bear witness to it — [22] the righteousness of God through faith in Jesus Christ for all who believe.
(Romans 3:21–22, ESV)

The Old Testament’s Law and Prophets point to Jesus Christ, the Savior, through whom the Law has been fulfilled for humanity. Paul states there is no distinction between Jews and Gentiles, for all have the same Lord (Rom 10:12). The work of Christ saves both the Gentiles and the Jews. Because the purpose of the law is not to justify but to condemn transgressors, “For by works of the law no human being will be justified in his sight, since through the law comes knowledge of sin” (Rom 3:20). The Bible does not teach Old Testament’s Israelites were saved by the Law; instead, it shows how utterly impossible for man to attain salvation by doing the work of the Law — he must place his hope and trust in the promised Messiah.

Why then the law? It was added because of transgressions, until the offspring should come to whom the promise had been made, and it was put in place through angels by an intermediary.
(Gal 3:19)

When sin enters humanity, we become enslaved and powerless to escape its tyranny. The Law was given to God’s chosen people, Israel, as a reminder of God’s standard of righteousness to give them hope that a promised Messiah who will fulfill “what the law could not do for it is weak through the flesh” (Rom 8:3). God’s Protevangelium promised the Offspring of the woman would cause a blow to the serpent’s head (Gen 3:15) nullifying its venomous sting against God’s chosen people. The purpose of the Law, then, is to point to the Messiah, who would come and fulfill its demands completely.

Wherefore the law was our schoolmaster to bring us unto Christ, that we might be justified by faith.
(Gal 3:24)

Thus, because of sin, humanity (both Jews and Gentiles) cannot fulfill the Law’s demands, which leads them to the point of despair in themselves and cries to the Lord for a Redeemer.

The intent of the Law in the Old Testament showed us the impossible standard of righteousness unattainable by sinful flesh. Confined in the nature of sin, the Law magnifies and intensifies the presence of sin (Rom 7:13) and drives even the Apostle Paul to the point of crying out, “O wretched man that I am! Who shall deliver me from the body of this death” (Rom 7:24)? The Law cannot save (Acts 13:39), and thus obedience to the Law is impossible because the “carnal mind is enmity against God: for it is not subject to the law of God, neither indeed can be” (Rom 8:7). So then, the flesh cannot please God in its sinful condition; only the spotless Lamb of God can fulfill the Law’s demands. At the cross, our Lord was crucified, not for His sins, but for the sins of the world, not only to save those alive but for those who died in their hope for the Messiah (Heb 11:13) and those who would be born. The Old Testament does not teach that the Law will save those who obey its commands but points to the One who saves Jews and Gentiles — all those who call on the name of Jesus will be Saved.