What does it mean to offer yourself as a living sacrifice?
In the context of Romans 12, I think it means that we are to sacrifice our sense of self-importance for the sake of serving the brethren in Christ.
Our natural mind can be one of self-preservation to the exclusion of caring for the needs of others. Our natural mind can create unnecessary hierarchies of importance, thereby esteeming ourselves or some too highly. Our natural mind can become preoccupied with using our abilities to serve ourselves and prosper in the world to the exclusion of using our abilities to serve others.
Sacrificing our self-importance places us in a position of usefulness for God and for our brethren. Having received the forgiveness of sins that was bought with the blood of Jesus, it is our reasonable service to shift our focus from one of self-importance to one that prioritizes the needs of others–especially among fellow believers.
To be holy is to be set apart, not for our use, but for God’s. This does not mean we do not struggle with sin. However, we should not sin. Only Jesus is truly holy in the perfect sense, but by faith in Him we can learn to lay aside sin so that we do not serve self, but others as Jesus would have us to do.
Sacrificing self can also mean laying our lives down in a literal sense for the sake of the gospel and in faithfulness to Jesus. However, that would most likely prove very difficult to do if we cannot first sacrifice our self in daily, practical ways. “If you do not love your brother that you see, how can you love God who you cannot see?”
I beseech you therefore, brethren, by the mercies of God, that ye present your bodies a living sacrifice, holy, acceptable unto God, which is your reasonable service.
2 And be not conformed to this world: but be ye transformed by the renewing of your mind, that ye may prove what is that good, and acceptable, and perfect, will of God.
3 For I say, through the grace given unto me, to every man that is among you, not to think of himself more highly than he ought to think; but to think soberly, according as God hath dealt to every man the measure of faith.
4 For as we have many members in one body, and all members have not the same office:
5 So we, being many, are one body in Christ, and every one members one of another.
6 Having then gifts differing according to the grace that is given to us, whether prophecy, let us prophesy according to the proportion of faith;
7 Or ministry, let us wait on our ministering: or he that teacheth, on teaching;
8 Or he that exhorteth, on exhortation: he that giveth, let him do it with simplicity; he that ruleth, with diligence; he that sheweth mercy, with cheerfulness.
9 Let love be without dissimulation. Abhor that which is evil; cleave to that which is good.
10 Be kindly affectioned one to another with brotherly love; in honour preferring one another;
11 Not slothful in business; fervent in spirit; serving the Lord;
12 Rejoicing in hope; patient in tribulation; continuing instant in prayer;
13 Distributing to the necessity of saints; given to hospitality.
14 Bless them which persecute you: bless, and curse not.
15 Rejoice with them that do rejoice, and weep with them that weep.
16 Be of the same mind one toward another. Mind not high things, but condescend to men of low estate. Be not wise in your own conceits.
17 Recompense to no man evil for evil. Provide things honest in the sight of all men.
18 If it be possible, as much as lieth in you, live peaceably with all men.
19 Dearly beloved, avenge not yourselves, but rather give place unto wrath: for it is written, Vengeance is mine; I will repay, saith the Lord.
20 Therefore if thine enemy hunger, feed him; if he thirst, give him drink: for in so doing thou shalt heap coals of fire on his head.
21 Be not overcome of evil, but overcome evil with good.