The Old Covenant which God made with Israel at Sinai contained a Law which, if kept, would give eternal life.
Leviticus 18:5 Ye shall therefore keep my statutes, and my judgments: which if a man do, he shall live in them: I am the LORD.
Ezekiel 20:11 And I gave them my statutes, and shewed them my judgments, which if a man do, he shall even live in them.
Galatians 3:12 And the law is not of faith: but, The man that doeth them shall live in them.
The problem was that none (save Jesus Christ) could keep it.
This meant that the Law served to condemn men and women to death.
Deuteronomy 27:26 Cursed be he that confirmeth not all the words of this law to do them. And all the people shall say, Amen.
Galatians 3:10 For as many as are of the works of the law are under the curse: for it is written, Cursed is every one that continueth not in all things which are written in the book of the law to do them.
Romans 7:10 And the commandment, which was ordained to life, I found to be unto death.
The Law of Moses was effectively a ministration of death.
2 Corinthians 3:7 But if the ministration of death, written and engraven in stones, was glorious, so that the children of Israel could not stedfastly behold the face of Moses for the glory of his countenance; which glory was to be done away:
It made clear the necessity of a redeemer.
Thus, the need for a new ministration based not on the Law but on faith in the redeemer Jesus Christ.
God had promised a New Covenant
Jeremiah 31:31 Behold, the days come, saith the LORD, that I will make a new covenant with the house of Israel, and with the house of Judah: 32 Not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which my covenant they brake, although I was an husband unto them, saith the LORD: 33 But this shall be the covenant that I will make with the house of Israel; After those days, saith the LORD, I will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people. 34 And they shall teach no more every man his neighbour, and every man his brother, saying, Know the LORD: for they shall all know me, from the least of them unto the greatest of them, saith the LORD: for I will forgive their iniquity, and I will remember their sin no more.
Hebrews 8:8 For finding fault with them, he saith, Behold, the days come, saith the Lord, when I will make a new covenant with the house of Israel and with the house of Judah… 13 In that he saith, A new covenant, he hath made the first old. Now that which decayeth and waxeth old is ready to vanish away.
Jesus Christ inaugurated a New Covenant, ratified with his own blood.
Hebrews 12:24 And to Jesus the mediator of the new covenant, and to the blood of sprinkling, that speaketh better things than that of Abel.
This New Covenant is termed the ministration of the Spirit.
2 Corinthians 3:8 How shall not the ministration of the spirit be rather glorious? 9 For if the ministration of condemnation be glory, much more doth the ministration of righteousness exceed in glory. 10 For even that which was made glorious had no glory in this respect, by reason of the glory that excelleth. 11 For if that which is done away was glorious, much more that which remaineth is glorious.
The New Covenant is a ministration of life: for by believing and obeying the gospel preached by Jesus Christ we can have forgiveness of sins and the gift of eternal life when he returns to raise and judge the dead and establish the kingdom of God.
I hope you find this helpful.