"I eat meat and oil."
Translation:Ben et ve yağ yerim.
I personally disagree with that last point (I say bread and oil), but what is important to know is that this sentence is just a nonsense sentence. Don't stress over it making perfect sense. We also have birds that read, people who own elephants and multiple houses, and I am even an owl.
"Eti" is the accusative case, and you use it for direct objects, but only for SPECIFIC direct objects (where we use "the" in English). If you wrote this sentence with accusative endings ("ben eti ve yağı yerim") it would mean, "I eat THE meat and THE oil" (instead of just, "I eat meat and oil").
That would be "I eat the meat and the oil." Adding the accusative endings to the direct objects (et + i and yağ + ı) makes those nouns specific direct objects, which means that in English, they would have to use the article "the."
Since the sentence here is "I eat meat and oil" (no "the" with the direct objects), these nouns are general direct objects, and don't take the accusative ending in Turkish.