We need to understand that what Paul is writing - "servants obey in all things" - excludes breaking the commands of God. He means obey all things pertaining to one's occupation.
Slavery was an accepted practice in Roman culture. Everybody who was anybody had slaves. Rome's population (just the city of Rome) has been estimated at well over one million people during the time that this book was written. One-half of the people in the city were slaves! And they were not, in most cases, just menial workers; slavery extended into the professions. In those days, doctors were often slaves, as were schoolteachers. Slavery extended into every area of society.
Were the apostles social crusaders? No, they were not. They did not try to change society. Their job was to work on changing individuals, especially those within the church. God permitted slavery to exist, and through Paul, He told Christians to operate within it. Not to overthrow it, but to work within it. Nobody is saying that the Bible says slavery is good. The Bible does not say such a thing. God wants everyone to be free. In this case, though, slavery was a part of the culture, and God nowhere instructs His people to overthrow it.
Today, very few people have ever been a slave like those whom Paul was addressing here. However, most of us work for a living, and the principle holds true for that area.