Place drop cloths on the floor and the countertop around the cabinets, and tape the cloths in place. Unscrew the cabinet hinges from the wall, and remove them from the wall.
Open any windows in the room, and put on gloves and goggles.
Apply varnish remover to the cabinets with a brush in the direction of the wood’s grain, and allow it to sit until the finish of the wood crinkles. Scrape the finish off with the plastic scrapper.
Wipe the cabinets down with mineral spirits on a cloth to remove any varnish remover still on the wood. Let the wood dry for 24 to 48 hours.
Sand the cabinets with a 120-grit sandpaper in the direction of the oak’s grain to smooth out any lumps in the wood. Sand the cabinets again with 220-grit sandpaper to smooth out all the wood on the cabinets. Wipe the dust off the cabinets with a damp cloth.
Apply the wood filler in a color that matches the stain you will finish the oak cabinets with. Use a cloth to apply the filler evenly to the surface of the cabinets. Wipe off any excess filler with the plastic scrapper. Sand the wood lightly with 220-grit sandpaper to even out the filler. Wipe the wood down with a damp cloth.
Wipe on stain with a cloth, and coat the oak cabinets evenly. Allow the stain to soak in for 15 to 20 minutes to hide the grain. Wipe off any excess stain. Repeat the process if you want darker cabinets.
Replace the cabinets, and screw their hinges into place after the stain is dry.