Hebrews 13:10-14 meaning

The Pauline Author just referenced the insufficiency of the Old Covenant and now contrasts the old system with the new. He says we have an altar from which those who serve the tabernacle have no right to eat (v 10). We refers to believers who have trusted Christ and are saved from hell to heaven. The altar symbolizes the sacrifice of Christ which allows us to be justified before God. Those who depend on the tabernacle (the Old Covenant) or who rely on rules and regulations to justify them, do not get to eat—that is, to share in the benefits that Christ's sacrifice provides. 

We have gained justification and forgiveness because of what Christ has done for us, and no set of religious rules or regulations can do the same. They will only serve to rob us of our freedom.

Now the Pauline Author compares the old sacrifices with that of Christ: For the bodies of those animals whose blood is brought into the holy place by the high priest as an offering for sin, are burned outside the camp (v 11). Under the Old Covenant, the Day of Atonement happened once a year. The high priest would make sacrifices on behalf of the people and sprinkle the blood on the mercy seat of the ark of the covenant. 

This act made atonement for all the unrepented sins of the Israelites. The blood of the sacrifices was brought into the holy place in the temple, but the carcasses of the animals were taken and burned outside the camp since they were considered unclean (Leviticus 16:27). 

Jesus was treated in a similar fashion. Therefore Jesus also, that He might sanctify the people through His own blood, suffered outside the gate (v 12). Although His blood was able to sanctify the people, he was taken outside of Jerusalem and made to suffer outside the gate on a cross. He was crucified along the road outside the walls of Jerusalem. The atonement sacrifice was a part of the picture looking forward to Jesus.

The Pauline Author exhorts his readers: let us go out to Him outside the camp,  bearing His reproach (v 13). He asks us to be willing to be excluded, rejected by the world system. Instead of being afraid of what men can do to us or getting distracted by the pleasures of this world, we are called to follow God and to endure until the end and finish ("teleo"). Hebrews 11 gave us many examples of men and women who chose to bear contempt from the world in order to obey God. We are called to follow God even when it means leaving what is comfortable or acceptable according to others.

Bearing rejection and persecution from others can tempt us to turn away from God and abandon our faith, but as we saw in Hebrews 11, men and women of faith were able to bear terrible things because they knew that they were strangers in this land. The Author writes, For here we do not have a lasting city, but we are seeking the city which is to come (v 14). 

Our true home is the city which is to come. If we endure shame and follow Christ's example, we have an opportunity to reign with Christ in His new city (Hebrews 12:28). We are willing to go outside of the gate because this is not our true home, and we have a hope of receiving a great inheritance with Christ if we remain faithful.