The Sun isn't "made of fire". It's made mostly of hydrogen and helium. Its heat and light come from nuclear fusion, a very different process that doesn't require oxygen. Ordinary fire is a chemical reaction; fusion merges hydrogen nuclei into helium, and produces much more energy.
How sun is burning without oxygen?
So, the Sun can "burn" hydrogen to helium without the need for oxygen. It should be noted that in the presence of carbon, nitrogen and oxygen, stars heavier than the Sun may burn hydrogen to helium by using the C, N and O as catalysts. Even in these stars, however, an absence of oxygen does not prevent nuclear burning.
How is the sun able to sustain itself?
Because it is so massive, it has immense gravity, enough gravitational force to hold all of that hydrogen and helium together (and to hold all of the planets in their orbits around the sun). We say the sun burns, but it doesn't burn like wood burns. Instead, the sun is a gigantic nuclear reactor.