All life on earth is powered by the sun. We get our energy by eating plants, which are directly powered by the sun through photosynthesis, and animals, which are indirectly powered by the sun by eating plants. We are absolutely 100% dependent on the sun and we would not survive very long at all without it.
Also I want to disagree with your statement that the sun’s continuous fuel supply is a mystery. We seem to have a pretty good grasp on it.
The sun is powered by a nuclear fusion chain reaction called the proton-proton chain reaction. Two hydrogen atoms fuse, generating deuterium, a heavy isotope of hydrogen (and some other byproducts). Deuterium can then fuse with another hydrogen atom to form 3He, a light helium isotope. From there, the reaction can proceed along one of four paths to form normal helium. One of the byproducts of each step of the reaction is the release of energy, yielding a net of about 27MeV. The sheer number of times this happens each second are what generate the massive amount of energy the sun gives off. Wikipedia says it better than I can:
The proton-proton chain occurs around 9.2 × 1037 times each second in the core of the Sun. Since this reaction uses four protons, it converts about 3.7 × 1038 protons (hydrogen nuclei) to helium nuclei every second (out of a total of ~8.9 × 1056 free protons in the Sun), or about 6.2 × 1011 kg per second.[36] Since fusing hydrogen into helium releases around 0.7% of the fused mass as energy,[37] the Sun releases energy at the matter–energy conversion rate of 4.26 million metric tons per second, 383 yottawatts (3.83×1026 W),[36] or 9.15 × 1010 megatons of TNT per second.
So why doesn’t it run out? Because the sun is just that big. And we understand this so well that we can predict roughly when it will run out of fuel and what will happen to the sun then. Again, borrowed from wikipedia:
The Sun is about halfway through its main-sequence evolution, during which nuclear fusion reactions in its core fuse hydrogen into helium. Each second, more than four million metric tons of matter are converted into energy within the Sun’s core, producing neutrinos and solar radiation. At this rate, the Sun has so far converted around 100 Earth-masses of matter into energy. The Sun will spend a total of approximately 10 billion years as a main sequence star.[87]
The Sun does not have enough mass to explode as a supernova. Instead, in about 5 billion years, it will enter a red giant phase, its outer layers expanding as the hydrogen fuel in the core is consumed and the core contracts and heats up. Helium fusion will begin when the core temperature reaches around 100 million kelvins and will produce carbon, entering the asymptotic giant branch phase.[26]
So yes, someday the sun will run out. But not for a very long time.
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