For this achievement you need to complete 72 Hour mode followed by Overtime Mode without being knocked out by either the Cultists or the Special Forces.
The Cultists spawn during Day 2, and their arrival is preceded by purple flyers appearing around the Mall. Once this occurs, step into Paradise Plaza and a cutscene will play. The Cultists are yellow raincoat wearing jerks who stand around in groups of 10 or so and creep slowly towards you, before lunging at you with a knife. Their killer attack is the dust attack, which will catch you by surprise. If that hits, you'll hit a loading screen, and wake up in a box in a room in Leisure Park, stripped naked and devoid of any items. If this happens, your Unbreakable run is void.
To prevent this try engaging the Cultists from afar, preferably using the Real Mega Buster, just to be on the safe side.
The Special Forces appear later in 72 Hour Mode, and are present in Overtime Mode as well. A cutscene will play showing them rappelling into Entrance Plaza during Day 4. If you die while the Special Forces are in the mall, instead of having to reload a save you will wake up tied up in a Special Forces helicopter. If that happens, your Unbreakable run is also void.
To prevent this happening, rush the soldiers with a hard-hitting melee weapon like the Small Chainsaw and make sure to stock up on healing items (such as the infinite Orange Juice in Colombian Roastmasters in Paradise Plaza).