Albert Barnes on Matthew 10:38

And he that taketh not his cross, etc. When persons were condemned to be crucified, a part of the sentence was, that they should carry the cross on which they were to die to the place of execution. Thus Christ carried his, till he fainted from fatigue and exhaustion. Mt 27:31. The cross was usually composed of two rough beams of wood, united in the form of this figure. It was an instrument of death. Mt 27:31,32. To carry it was burdensome, was disgraceful, was trying to the feelings, was an addition to the punishment. So, to carry the cross is a figurative expression, denoting that we must endure whatever is burdensome, or trying, or considered as disgraceful, in following Christ. It consists simply in doing our duty, let the world think of it or speak of it as they may. It does not consist in making trouble for ourselves, or doing things merely to be opposed; it is doing just what is required of us in the Scriptures, let it produce whatever shame, disgrace, or pain it may. This every follower of Jesus is required to do.