"How do cells eliminate waste?"
Cells need to "recycle" different compartments that are called "organelles", or specialized structures within living cells. To do this, they bank on an organelle named the lysosome, which is basically the cellular stomach.
A cell takes in food and water through the cell membrane, as well as eliminates waste through the cell membrane.
I hope this answers your question.. If not, I don't mind explaining some more.