I have a shop cabinet (poplar, if it matters) that had a grease gun sitting on top of it in storage. Unfortunately the grease gun leaked and left a huge streak down the door and side.

Can I paint this thing somehow? I can't imagine I'll be able to clean all the grease that soaked into the wood. There's nothing on the surface right now but it definitely soaked in, as the wood was unfinished. If I clean it as best as I can, is there a primer that'll stick to this, or am I doomed to leave it unpainted?