Will not start?

The description could be a faulty fuel pump. Try this, remove the fuel cap, then have someone just turn the key to run, you should hear the pump in the tank turn on for 2 seconds and then off.
If you don't hear it then there are a couple more tests to try before condemning the pump. One is a fuel pressure test. You should see right around 60 psi with the key on engine off.

https://www.2carpros.com/articles/how-to-check-fuel-system-pressure-and-regulator

The last test is simpler, open the air cleaner so you can get to the air intake and spray some starting fluid into the duct, try starting it, if it suddenly starts or at least tries to start you have a bad pump.

https://www.2carpros.com/articles/how-to-replace-an-electric-fuel-pump

You need to drop the tank to get to the pump in that vehicle.
However, I would also use the pressure test hose to pull a sample of the gas out of the fuel system. Especially as this started not long after you visited the station. Turn the key on and off a few times to catch the fuel in a clear glass container. Set it on a shelf and let it set for 20 minutes or so. Now look at it. If you have water in the fuel, it will settle out in a layer under the gas. If that is the case then you would use the in tank pump to pump all the fuel out, then remove the tank and remove the pump and clean any residue out of the tank and replace the pump assembly. Then refill with known good fuel.

The symptoms don't fit with a timing belt. They don't start randomly shutting off, they just stop running the second the belt breaks. If I remember you can remove the oil fill cap and see the cam, look in there while someone turns the key, if the cam moves it's not the belt.

Some of those point to a timing issue but there are others that point toward fuel. It's possible you got hit with two problems at once. The engine started misfiring due to a lean condition and the shocks through the crankshaft caused a timing belt failure that broke when the vehicle was being started. I would look at the cams first. If it is a failed belt, then I would suggest based on that and the mileage that either a lower mileage used or a remanufactured engine might be better than just replacing the belt.

With the mileage you have on it I would probably pull the engine down and rebuild it. The 2.4 is known to start using oil from worn rings on the turbo versions the seals are usually starting to fail. The engine is a non-interference design so valve damage from a bad belt usually isn't an issue. If it's only a bad belt you would want to replace the belt, the tensioner and idler as well as the cam and crank sprockets due to wear.