So, what's the short answer? Well, yes, homemade caramel does need to be refrigerated - but only if you want it to last for more than a few days. If you're planning on using it up quickly, you can probably get away with storing it at room temperature in an airtight container.
But, let's be real - who makes caramel and then uses it all up in one day? It's like buying a gallon of ice cream and eating it all in one sitting (although, no judgment if you've been there too). The point is, if you want your caramel to last, you need to refrigerate it.