I'm sure you will have many idea/suggestions for swarf containment at the lathe. But I'd also suggest in addition a true heavy cotton duck (or better) painter's cloth on the rug. After all it protects from paint.
But plastic tarps are to be avoided at all costs. They are designed to keep paint from drying, and put it all back on your shoes when you walk, unlike cloth.
Likewise oil and swarf. Cloth is heavy and stays put. Plastic tarps love to bunch up and are slippery.
A false floor (a panel of plywood) over the carpet might also serve well -- or over the cloth tarp, for good measure.
A change of slip-on shoes in your work area will help keep swarf from tracking through the house, too.