I think "solving that question" means successfully finding somewhere to live besides days. So "solving" just means finding the answer or answers to. In math, solving x squared equals nine means figuring out that x equals three or minus three.
What might the answers be? Maybe living at night, or at random times, perhaps ignoring time completely, coming late or early for doctor's appointments, going to church at night, and/or going generally mad?
Though it could mean death, as pointed out in at least one prior answer, I think it could also mean (or allude to) some sort of insanity or evil or nonconformism so extreme that one is thought to be in need of help and/or management.
The phrase "in their long coats", suggests to me "men in white coats", and the idea that the doctor in his white coat and the priest in his black coat are coming to apply pressure to the person to live a more typical life.
The priest and the doctor are running over the fields. It suggests some kind of emergency that they are running, which could be that someone is dying, but also that someone is getting out of hand, though not so much that the police or army is needed. But I do wonder why "over the fields". A doctor or priest would normally come by road, surely? What could this be alluding to? Maybe doctor and priest means doctor, priest, policeman, soldier, and every other representative of society that can be called upon to get a person to act more normal.