When cleaning your weapons on a surface on which you don’t want to risk making a mess (say, the dining room table), use an adult-sized absorbent bed pad, the kind they use in home health care (e.g., Chux) on your work surface.
The pad absorbs any oil (or other debris) you may drip, the white surface makes it easy to see any parts or tools you lay down, the plastic backing keeps liquids from seeping through to the surface below, and when you’re done you just wad it up and toss it.
They’re available in most any drug store, grocery store, or megamarket, usually wherever they stock the adult diapers. If buying them locally is too embarrassing, you can always order them on line.