Make root beer without a refrigerator

There are a few alternative methods:

• You can force carbonate by using a CO2 tank. This may be expensive if you don't already have the equipment. (I make soda, but only because I had the equipment from beer brewing!)
• Using potassium sorbate to stop the yeast propagation (but not fermentation - existing yeast will still continue to ferment.) Apart from the lack of control, that would mean opening each bottle and adding something, which might cause the soda to foam over and/or reduce the level of carbonation.
• Heating to pasteurize - most yeast will not survive above 40-45C, so raising the temperature to that point may help stall fermentation. However, this will also greatly increase the CO2 pressure in the bottles, possibly above their breaking point. Only try this with plastic bottles.
• A final option is using artificial sweeteners. Some artificial sweeteners are unfermentable, such as Stevia. You use the artificial sweetener for the sweet taste, and just enough real sugar to get carbonation (about 10 grams per liter). I have not tried this so can't tell you how it will taste. The yeast will ferment the real sugar, to make CO2, but will not ferment the stevia. You can use 1/4 as much yeast as given in the recipe when trying this method.

As you can see, putting the bottles in the fridge or storing in a cool cellar is the simplest option!

When making soda, I would advise against using glass bottles at least for the first couple of times, since the co2 pressure can become very high, and can lead to exploding bottles.