I am developing a fictional world. There has been no animal life for 500 years. Assuming that humans were technologically advanced enough to keep the environment stable, what would they eat?
I don't really see any reason to close this question, but the answer is pretty trivial. What do you eat that isn't animals? Here is a list of foods.
If you are more worried about the ecosystem effect of not having animals (such as pollinators), here is a list of bee-pollinated edible plants. There are some important foods on that list, but plenty of important plants that are not.
A full agricultural complex will have starchy staples, legumes, oil seeds, fiber plants, fruits (for vitamin C), and leafy greens. Examples in each category that are wind pollinated include:
Starchy staples: Wheat, rice, corn and bananas.
Legumes: Soybeans, lentil, pea, peanut
Oils: Soybeans, peanut, oil palm, olive
Fiber: Jute, hemp, raffia. Also, you can just replace cotton with nylon or polyester and skip the plants.
Fruits: Banana, dates, heirloom tomatoes, pineapple
Leafy greens: These will grow leaves without pollination; seeds can be produced from human pollinated hybrids.
So even without pollinators, you will be able to eat.
There is a long history of vegetarian / vegan lifestyles (ranging back to the former 7th centruy BCE https://en.m.wikipedia.org/wiki/Vegetarianism and the latter 1944 https://en.m.wikipedia.org/wiki/Veganism). There are indicating that not eating meat or any animal based foods is survivable -- perhaps even more "healthy."
We eat protein to rebuild and repair our bodies and if necessary act as fuel, though the fuel aspect is better served by carbohydrates. One thing to keep in mind is that protein (one of the main things you get from food outside fibre, vitamins/minerals) is made up of amino acids. Amino acids are essentially complex molecules, found in cells, with short lifetimes. When you stop eating a diverse of foods your body must remanufacture the amino acids or operate in a deficient manner (this is a tree top level discussion, for instance animal fat can be considered "high powered" fuel -- forgive the lack of references, I've worked professionally with biology types).
All of this being said, you can produce animal proteins artificially in a lab. This would suggest that it could be done on an industrial level. I can't find any examples of anyone pursuing this strategy (again wikipedia https://en.m.wikipedia.org/wiki/Protein).
They could be vegans, but you know the truth...
Edit: Per Vincent's excellent comment below, not everyone has seen the movie Soylent Green (but you should fix that if you have not). The premise is that in a post-apocalyptic/Utopian future society people often opt to commit a form of assisted suicide where they are drugged into a state of euphoria before being put to sleep. Also, completely unrelated, everyone eats this super health food called Soylent Green.
Twist! in the words of Charlton Heston, "Soylent Green is people!!!!"
Without non-human animals, plants would need a source of carbon dioxide. When it comes to gases, plants are net oxygen producers. Fungi, on the other hand, are net carbon dioxide producers, just as animals are. The elimination of animal life might create more niches for fungal life.
It's possible that humans could selectively breed the fungi for nutritional value. Fungal steak, anyone?