Neighbour Margaret Graham told The Daily Telegraph that Cheryl "said her husband was down in the cellar trying to fix a generator".
"She went down to see what was going on, found him unconscious and she had to come up herself because the fumes were too strong," Ms Graham said.
"Then her sons came on the scene and the boys went down to help their dad and they got trapped too."
The crews used breathing apparatuses to enter the shed and retrieved the three men, despite "difficult access", just before 5pm.
NSW Ambulance Inspector Lisa Darley said paramedics "found a very distressed wife" when they arrived on the scene.
"We could see that there were three patients inside before we were overwhelmed by the fumes … and we had to wait for the NSW Fire & Rescue to come with their specialised equipment," she said.
A NSW Ambulance spokesman who described the tragedy as "a hazardous material incident" said all three men went into cardiac arrest.
"All three were taken to Broken Hill Hospital but sadly passed away," police said in a statement.
A NSW Police spokeswoman earlier told news.com.au the deaths were not suspicious, no one was being sought in relation to the matter, and "it was not suicide".
Officers from Barrier Police District have established a crime scene at the premises but have not yet been able to "gain access safely to conduct a full investigation".
Police inquiries are continuing and a report will be prepared for the coroner.
A similar situation occurred on a farm in the NSW Southern Tablelands in February 2017 when a husband, wife and their neighbour — all in their 60s — died in an empty water tank.