Scrub the Brazilian eucalyptus with water and mild dishwashing soap to remove all traces of surface dirt. Dip a sponge into the soapy mixture. Scrub the wood with the sponge and rinse with clean water.
Attach medium grit sandpaper to palm sander. Sand the entire piece of furniture until the original color has returned. Quit sanding when all traces of weathered gray color are gone.
Remove the medium grit paper and attach fine grit sandpaper to the palm sander. Smooth out any imperfections in the wood. The surface should be soft and smooth under your bare hand. Wipe away the sanding dust with a damp rag.
Dip a clean rag into tung oil. Rub the oil into the wood, leaving enough oil on the surface so the furniture looks damp. Let the wood absorb the oil overnight.
Apply a second coat of oil to the wood the same way you applied the first. Let the oil soak into the wood overnight.