The cheapest and safest way to heat a shed depends of the type of activities you intend to do in the shed. The basic options you have are similar to home heating. Fire, propane, kerosene and electric all have advantages to be considered.
Use fire for the most basic and cheapest form of heat. If you do any woodworking, you will find a fire to be a good use of scrap materials. A drum stove can provide enough heat for an average sized home shed. You can build one on your own using a 55-gallon drum or even an old hot water heater. Instructions can be found on a number of websites for either stove. Similarly, you can choose to install a small fireplace along a wall or in a corner, but be aware that a chimney is necessary.
The open flame can prove dangerous if you do not take the proper safety precautions. You should never leave a fire unattended, even for a second. The flame will also release CO2 into the air, so you must properly vent the drum or fireplace to lessen the dangers associated with inhalation. For small sheds or ones that are old and dry, it is best to avoid an open flame when possible.
Perhaps the most convenient and least risky way to heat a shed is with a small electric fan heater or portable electric space heater. These don't take long to heat up and will provide the quickest and easiest heat with the least hassle and very low up front costs. Unlike most other sources of cheap heat, ventilation is not a concern with these small heaters.
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Propane and Kerosene
Use propane and kerosene powered heaters if you like the idea of a space heater, but have a shed with limited electrical capacity. These small heaters can provide warmth quicker and in greater concentration than many small electric heaters.
A kerosene or propane heater requires little maintenance, although it is absolutely necessary for you to follow safety precautions. These heaters have an open flame, so you must ventilate them just as you would with a wood fire. Although maintenance on the heater itself is minimal, there are many precautions that you must take when using fuel heaters. All of these safety issues can make them unappealing. For example, many heaters cannot be stored in the cold, and you should always refill tanks in an open area where leaks will not endanger your structure. All of the moving required for proper safety can make them more complicated to use than they first appear. As with any open flame, take caution with placement and be sure that there are no burnable items near the heater.
Fire is perhaps the most effective form of heat available for a shed, but it is most likely not the safest option. With the danger of igniting materials along with continued inhalation of CO2, you may find that it is more complicated to run what appears to be the cheapest form of heat. If you would like to use fire, a drum stove or fireplace with a ventilated chimney is the cheapest the safest option when monitored properly.
Kerosene and propane may seem safer than a drum stove, but Consumer Reports studies have found that they release so much CO2 that they recommend them only for areas with significant ventilation or outdoors.
With any heating project, take into consideration your state and local safety codes and the recommendations of the heater's manufacturer.