ANSWER: The book of Romans, chapter 12, in the very first verse of the chapter, states what it means to be a true servant of the Eternal. The apostle Paul states that believers should present their bodies to God as a holy, acceptable, LIVING sacrifice to him. Doing this, Paul goes on to state, is our reasonable response to all that our Father had done for us.
There are a number of ways this question could be answered. To be brief, I will simply contrast the idea of being a living sacrifice to the Eternal with two other kinds. One kind is to a pagan false god and the other to God based on his Old Testament law.
In the non-Christian religion of the Roman Empire, some of the temples to pagan deities included such worship practices as temple prostitution. Paul makes a number of references to this in his epistles, mentioning how the Christian converts came out of such worthless practices. With this in mind, we can see that being an 'offering' to God would mean not taking part in the evil practices of pagan worship, which God hates, but rather doing those things which he wants us to do.
In the Old Testament, the priests of the Temple would offer animals as part of the worship of God, cutting them up and offering them on the altar. Paul does tells us, however, that God would prefer that WE become a sacrifice while alive, which is far greater. In fact, even under the Old Covenant God considered obedience to be far better and of greater value to him than any offering of animals (1Samuel 15:22 - 23, Psalm 40:6 - 8, Amos 5:21 - 24).
Our Father wants us to lay ourselves aside daily, dedicating ourselves to him and His way of life. The rest of Romans 12 gives us some examples of how we can sacrifice ourselves on a daily basis. Below are just a few of them.
We can sacrifice ourselves by not conforming to this evil world (Romans 12:2), by being transformed by a renewing of our minds, and by proving what is the good and acceptable and perfect will of God. We can also strive not to be vain and self-centered but in humility serve others with whatever gifts God we possess (verses 3 - 8), do not pretend to love but sincerely do so as Jesus loved us (verse 9), reject evil and hold fast to what is right and good, and finally to serve others based on THEIR needs not ours (verse 15).
In conclusion, rather than giving our physical life as a one-time offering (like an animal at Jerusalem's temple), a Christian is to offer themselves daily to serve God and love his fellow man. We do this not by being self-centered - doing things just to please our own wills - but by being focused on benefitting and blessing others. As a believer offers themselves in this way each day they become more and more like God - perfect - which is what our Father wants (Matthew 5:48). Thank you for your question about what does it mean to be a living sacrifice to God.