Plants, and some protoctists (algae, for example) use the sun's energy to fix carbon from the air (photosynthesis). Because the resulting compounds can release energy when they react with oxygen (or are otherwise broken down), animals can use these compounds inside their bodies to release energy for their own purposes.
Thus, the ultimate foundations of all food chains are plants, which is why they are sometimes called "producers" - they produce biomass using simple raw materials and the sun's energy.
You could eat algae or fungi (which includes yeast) which are neither plant, animal nor bacteria, but those are living things too...
Animals, fungi and bacteria all depend on plant (or protoctist) biomass for their energy needs, either by taking nourishment from plants directly, being parasitic on organisms that take nourishment from plants, or decomposing dead organisms that were nourished on plants.
Thus, it's not possible to get calories and other necessary nutrients without eating plants or eaters of plants, without eating living things. Photosynthesis is the only viable way to create material that can yield those nutrients.
Bacteria live inside us and on every living and non-living surface (unless sterilzed - by killing them) and in the air. We cannot avoid killing them, eating them or otherwise interacting with them.
So the answer to your question is no.
Some fruitarians try to eat in a way that minimises harm to plants, for example by only eating fruit that has fallen from the tree and so on, but such a limited diet would not be possible in the long term because of nutrient deficiencies (not to mention the impracticality of obtaining a constant supply of such foods).