How to determine how much heating oil is left in a tank like this if the gauge is broken?
You can use temperature strips purchased at a hardware store to show you where the oil level is. These strips change color with temperature, showing you where the cooler (or warmer) fuel oil is my measuring the temperature of the outside of the metal tank. If the room changes temperature throughout the day (like a garage), the strip will tell you where liquid resides inside the tank by indicating where the tank itself is heating/cooling more slowly than the ambient temperature.
Note that if the temperature of the room remains relatively constant (like a basement), you can pour a bit of hot water over the strip to indicate where the cooling effect of the heating oil is keeping the temperature of the tank from going up.
Propane tanks use this same principle, although the cooling effect of releasing pressurized propane gas doesn't really apply to your oil tank, but you can apply the same principle of temperature differences nonetheless.
If the length of the strip is an issue (i.e. you want to measure more than the "low mark" of the tank), you can always use thermochromc paints or films to create a temperature-sensitive strip over a longer length. Just be sure the pigments you select are designed to change within the temperature ranges you have in that room.
For a quick work around when the gauge is broken, no shopping trips needed, no tools needed, simply tap on the side of tank starting from the top and going down. The sound will change when you reach the approximate of the remaining fuel.
Also if you have a cap on top of the tank and enough clearance above the tank, you can use a long stick to determine the level, just like an oil dipstick.
Like before the user Jon has answered, I would recommend you highly make a new gauge. I would not play with knocking on the tank nor use this propane toys for this.
I think the big pipe (on the left side) is the input, right the output and in the middle is a cap (with of course i think the broken gauge)
If you have time to fix this problem, you can go to your local Hardware Store and buy a double meter stick made of plastic or better a Swedish model (made of wood). Also maybe buy a new cap, one that you can turn on simple with one hand. Then put the meter in the tank, mark it and cut it, so that you just can crap it close by hand at the top (of course the cap must be turn on also).
Then you have a good solution if you use this yearly. otherwise the effort is not worth. this is also the solution that real tanks must have.
I think this isn't' a good example of a lifehack question.
if you like you can mark the meter stick (calculate 1/4 full, 1/2, ect...)
Pour boiling water down the side of the tank.
There will be an obvious line where the heat is transferred away from the metal by the remaining fuel in the tank. The upper part, where there is no fuel, will dry quickly from the hot water heating the metal of the tank and staying hot. The lower area, where there is fuel, will remain wet longer, since the metal will be cooled by the fuel.
If the visual indicator is not as helpful as you'd like, you can also just feel the metal side of the tank, from the bottom, moving up, looking for the height at which the tank is noticeably warmer. That will mark the fuel level, as the fuel will cool the metal tank faster than the vapors above will.
Use a long stick or pole - tent poles are awesome. Insert straight down the fill hole, pull out and measure the length of wet on the stick. Measure your tank to find out your capacity and then, using your measurements Refer to this chart to determine your oil level: http://www.adamspetro.com/residential-heating-tank-chart
So, using the chart, if I had a 275H tank and I measured 10 inches on my stick then I would have about 96 gallons of fuel remaining.