In the Yeats poem, "The Ballad of Father Gilligan," the fact that half of his flock, half of his parishioners, are lying under green sod is responsible for the priest's exhausted state. The green sod refers to their graves.
Half of his flock is dead, and the father can't keep up. His job is to comfort the dying and administer the last rites. His people are dying so fast that he is exhausted. Even as he dozes off in a chair, he is summoned by another man dying.
The image of the green sod and the graves of his parishioners contributes to the plot of the narrative, explaining and echoing the death of the man who sends to him, as well as explaining why the priest falls asleep and fails to respond to the man's call. This sets up the intervention of God, who sends an angel to take the poor priest's place at the dying man's side.
Although in British English the word "sods" can be an obscenity, it also has a "decent" meaning. Sod in that context is an area of grass that also includes the dirt beneath it that is held together by the roots of the grass.
So, if part of his flock is under areas of grass like this, where would they be? People who are under the grass are dead people who have been buried.
So half of his flock lies under the green sods because they are dead. Gilligan is old and many of his people have died.
A widespread epidemic has devastated the Irish countryside and the poor village people have succumbed to it and lie buried under the ever green Irish countryside.
'Sod' means the layer of earth with vegetation on it. In the first stanza Yeats describes to us the weariness of Father Gilligan. He tells us that because of the epidemic almost half the village has died and lies buried "under the green sods" while the other half is lying in their sick beds. So, Father Gilligan the parish priest is kept busy day and night because he has to either conduct the funerals of his dying parishioners or administer communion to the remaining parishioners who are either sick or on the verge of death.
In the Bible the Priest is often referred to as a shepherd and his parishioners as his flock. So, Father Gilligan is equated with a shepherd who tends his parishioners who are his flock.
Ireland is an island and enjoys rain throughout the year and consequently the Irish countryside is always green. The color green is also the national color of Ireland and Ireland is picturesquely referred to as "the emerald isle."
The irony in this line is of course, that sheep usually graze on the green grass but here because of the devastating epidemic which has caused an ironic reversal, the sheep are lying buried under the "green sods."