Do what they believed in the 19th century: make your universe very young with a creator.
The problem of how did the Sun keep shining was one of the arguments that the Earth and Sun were very young. The size of the Sun had been estimated and no known fuel could keep the Sun shining for the millions or, haha, billions of years some of the radicals at the time were claiming. After all, they were geologists! They just categorized rocks.
As for the elements? God brought it all into existence about 10,000 years ago. Done. Don't believe in God? Then where did all this life come from? Atheism before Darwin was unscientific.
The cracks in this idea came from the then very new field of geology. The Industrial Revolution, and its great need for coal and mineral wealth, elevated geology from rock collecting to a systematic survey of the Earth. The geologists were finding more and more evidence that the Earth needed to be millions or billions of years old, but then nobody could explain how the Sun kept shining. Obviously the Earth couldn't be that old!
Then there's Darwinism, introduced in the late 1800s. Like geology, evolution needed millions or billions of years to do its thing, and this was time physics said the Earth did not have. Also, it didn't explain the origin of life, just the origin of the species.
Eventually the geologists and the Darwinists won, there was just too much evidence for an old Earth, but it doesn't have to be that way in your universe.
A good book to read on the subject is the pop sci book A Short History Of Nearly Everything by Bill Bryson (not to be confused with the metaphysics book A Brief History Of Everything). It's a history of the question "how old is the Earth" and our attempts to answer it. You can follow their chain of discovery and logic in realizing the Earth is very old and change some of the key discoveries to match a young Earth.