No man can serve two masters: for either he will hate the one, and love the other; or else he will hold to the one, and despise the other. Ye cannot serve God and mammon. Matt. 6:24.
That verse was undoubtedly more forceful in the ancient world than it is to us. The verb translated as "serve" is from doulos, the word for "slave." The Greek word behind "master" is kurios, denoting absolute ownership and nearly always rendered as "lord" in the New Testament. Thus the idea of Matthew 6:24 is that no one can be enslaved to two owners or lords at the same time.
To catch the full impact of this statement we need to realize that the ancient world did not regard a slave as a person but as a living tool. Slaves had no rights of their own. They were completely under the control of their masters, who could do with them as they wanted. Masters could sell slaves, beat them, throw them out, or even kill them.
A second thing to note is that in the ancient world slaves had no time of their own. All of it belonged to their master. In modern culture each worker has time off for personal needs. During that time they can have hobbies or even hold a second job. But that was not so in the ancient world of slavery. A slave's time belonged wholly to the master.
Jesus is saying that Christians must let God be the undisputed master of their lives. Paul makes the same point in Romans 6:16, in which he says we are slaves to either sin or righteousness, Satan or Christ.
Thus Christians, being enslaved to Jesus, always take God's will into consideration in all they do. They daily ask themselves, "What does God wish me to do?" Every moment of their time they live for Him. God has no part-time devotees who largely serve Him but them moonlight for some other master in their time off.
When Jesus said no person can serve two masters, He meant it.
Still, some of us try anyway, in spite of the impossibility. But in that attempt, whether we recognize it or not, we have really opted for Satan. "He who does not give himself wholly to God is under the control of another power, listening to another voice....Half-and-half service places the human agent on the side of the enemy as a successful ally of the host of darkness" (Thoughts From the Mount of Blessing, p. 94).