The Hebrew Bible does not teach that a sacrifice is the only way to be forgiven. There are verses that make it clear that God is able to forgive people when they have not offered a sacrifice.

Ezekiel 18:​21 But if the wicked will turn from all his sins that he has committed, and keep all my statutes, and do that which is lawful and right, he shall surely live, he shall not die. 22 None of his transgressions that he has committed shall be remembered against him; in his righteousness that he has done he shall live.

Isaiah 55:​7 Let the wicked forsake his way, and the unrighteous man his thoughts; and let him return to the LORD, and he will have mercy upon him; and to our God, for he will abundantly pardon.

Jonah 3:​10 And God saw their doings, that they turned from their evil way; and God repented of the evil, which he had said that he would do to them; and he did not do it.

The people of Nineveh repented by stopping the sins they were doing and by asking God to forgive them and that was enough for God to forgive them. Ezekiel and Isaiah taught that God will forgive the sinner who repents. None of these verses mention animal sacrifices. 

Christian apologists often teach that blood was the only way to make atonement in the Hebrew Bible. This is incorrect. Although there were animal sacrifices in the biblical law, they were never the only way to make atonement. Leviticus says that a poor person can offer flour instead of an animal sacrifice, this would not be possible if only blood could atone for sin. 

Leviticus 5:11 But if he is not able to bring two turtledoves, or two young pigeons, then he who sinned shall bring for his offering the tenth part of an ephah of fine flour for a sin offering; he shall put no oil upon it, nor shall he put any frankincense on it; for it is a sin offering.

Praying and asking God for forgiveness is the way to make atonement. 

Psalm 32:​5 I acknowledged my sin to you, and I did not hide my iniquity. I said, I will confess my transgressions to YHWH; and you forgave the iniquity of my sin. Selah.

The Psalmist confessed his sin directly to God and asked for forgiveness and that was enough for God to forgive his sin. This also proves that we do not need a mediator to talk to God. Anyone can talk to God and have their prayers answered if they are sincere. 

People were forgiven and still are forgiven without the sacrifices performed in the temple. God promised to have compassion on the Israelites when they repent even in exile, a situation where the temple did not exist and a sacrifice could not be offered: 

Deuteronomy 30:​1 And it shall come to pass, when all these things have come upon you, the blessing and the curse, which I have set before you, and you shall call them to mind among all the nations, where the Lord your God has driven you, ​2 And shall return to the Lord your God, and shall obey his voice according to all that I command you this day, you and your children, with all your heart, and with all your soul; ​3 That then the Lord your God will turn your captivity, and have compassion upon you, and will return and gather you from all the nations, where the Lord your God has scattered you. ​4 If your outcasts have been driven out to the farthest parts of heaven, from there will the Lord your God gather you, and from there will he fetch you….

All this proves that God is a forgiving God and that even though He can be angry with people, He can and does forgive people who are sincere.