Not all salvation talk within the Old Testament is regarding eternal salvation from Hell. This is the same situation with the New Testament. However, almost anything related to blood sacrifice within either testaments is absolutely about eternal salvation. Temporary remission of sins was provided until the ultimate sacrifice was provided. Those who died while in remission from observing the Law (animal sacrifices) are eternally saved.
Under the covenant promises, the thing that God required to be done was sacrifices being offered, confession of sins being made, and doing the commandments (see Matthew 28 and John 14). There were things that God told them to do in order to demonstrate their faith.
The Law in Deuteronomy 32:45-47 says that “doing the commandments is your life” (Proverbs 6:23) and that is very much different from what Paul states (“the Law is not your life and your life is in Christ” – Col 3:3-4) within his 13 epistles. The blood of the covenant was not applied to them until they, a member of the nation Israel, had agreed to perform all of the works of the Law. They had to agree to be obedient before the blood was applied. The works didn’t save them but the works were required for them to remain in the covenant.
[Exo 24:7-8 KJV] And he took the book of the covenant, and read in the audience of the people: and they said, All that the LORD hath said will we do, and be obedient. And Moses took the blood, and sprinkled it on the people, and said, Behold the blood of the covenant, which the LORD hath made with you concerning all these words.
Under the covenant program, under the covenant that God gave to Abraham in Genesis 17, there were things that needed to be done (circumcision, offering his son). Under the Law covenant (the Mosaic Covenant), there were more things that needed to be done (613 points of the Law). Under the covenant program, works were required:
[Lev 18:5 KJV] Ye shall therefore keep my statutes, and my judgments: which if a man do, he shall live in them: I am the LORD.
[Deut 32:46-47 KJV] And he said unto them, Set your hearts unto all the words which I testify among you this day, which ye shall command your children to observe to do, all the words of this law. For it is not a vain thing for you; because it is your life: and through this thing ye shall prolong your days in the land, whither ye go over Jordan to possess it.
[Luke 1:6 KJV] And they were both righteous before God, walking in all the commandments and ordinances of the Lord blameless.
[Matt 19:17 KJV] And he said unto him, Why callest thou me good? there is none good but one, that is, God: but if thou wilt enter into life, keep the commandments.
[Matt 21:43 KJV] Therefore say I unto you, The kingdom of God shall be taken from you, and given to a nation bringing forth the fruits thereof.
The covenant required works! Faith alone was not sufficient because their faith did not include the Gospel of the Grace of God, it did not include the preaching of the Cross, free justification as Paul preaches it in Romans 3. That faith was kept secret and was first revealed to Paul so that in this dispensation, grace and faith stand alone without works (Eph 2:8-9; Rom 4:5). This is very different from what happened under the covenant program with Israel.
The righteousness of the Law was by faith, but it was faith proven by works.