The survey of 14,000 local government workers found a "toxic cocktail" of declining staff numbers and increasing expectations from both employers and the public is piling on the pressure.
As many as 72% of those surveyed said stress is affecting how well they can do their jobs and 70% said workplace stress is affecting their personal life.
The union is calling on employers who are currently consulting on a pay offer to recognise that councils cannot function without their dedicated workforce, and to ease the pressure of the three-year pay freeze imposed on councils by central government.
It says that freezing pay while inflation remains high has left local government workers battling to make ends meet. Pay has fallen by 15% in real terms since the coalition took office, and only 40% of those surveyed by Unison now feel that they are paid fairly for the work that they do.
Heather Wakefield, head of local government at Unison, said: "Working in local government is like living in a pressure cooker and eventually the lid will blow off. Workers can't take any more.
"Multi-billion-pound cuts, and 250,000 job losses as calls for services increase, mean that impossible demands are being placed on stressed-out council workers.
"And the stress at home continues. The pay freeze means it's a constant financial juggling act as red bills pile in and wages just don't match up.
Wakefield added: "The Government has to ease the pressure on councils, allow them to pay staff a rise this year and slow the cuts and closures to give the public the services they need in this hour of need."