The reason for calling the elders was apparently because the sickness referred to by James was a sickness that resulted from sin.
The elders were the spiritual leaders of the assembly of believers, and because of the type of problem involved, they--not a physician--were to be called.
It was the responsibility of the spiritual leaders to deal with, and pray for, those who had gone astray and as a result had been stricken by a sickness.
James did not use the word that is associated with ceremonial anointing but the word that is associated with the treatment of wounds. The word James used is often found in secular medical treatises of New Testament times.
The oil was, in itself, a healing ointment. So we see that James was referring to the best-known medical treatment of the time; that is, rubbing with oil.
The rubbing with oil was to be accompanied by prayer. This was apparently to be done first because the original language indicates that the elders were to pray over the sick person, having anointed him with oil (see James 5:14).
From this passage some derive the teaching of divine healing apart from medicine, but such a view is not supported by the text.
The oil was an accepted medical treatment of the day, so this passage actually encourages the use of known medical practices in addition to prayer for healing. Of course, the confidence of the elders was to be in God's ability to heal, not in the medical treatment itself.
"For if we would judge ourselves, we should not be judged" (1 Cor. 11:31).