Musty smells usually mean mold or mildew, which are two words for the same thing. Several thousand mold species exist, and some are benign while some, such as stachybotrys, or black mold, can be harmful. They all have one thing in common, though, and that is the fact that they need moisture to survive. Consequently, the characteristic moldy smell coming from your oak furniture is a sign of high humidity, condensation or poor ventilation -- or all three. To get rid of the smell, it's imperative to dry the furniture and physically remove the spores. Disinfecting and sealing the wood also help.
Add 1/2 cup of laundry detergent or general-purpose cleaning detergent to a gallon of water and wash the musty oak furniture with a sponge to remove the mold spores. You don't need bleach -- it won't penetrate the pores of the wood and kill the roots, and it can damage the finish.
Wash the wood with a solution consisting of 1 cup of vinegar per gallon of water -- it's a better way to disinfect than washing with bleach. Vinegar is acidic and soaks into the pores, but it won't kill all the spores -- nothing will. Vinegar can dull the existing finish, so avoid it if you don't plan to apply more finish.
Dry the furniture thoroughly, either by using a rag or -- preferably -- by taking it outside and leaving it in the sun. Dry air and ventilation will evaporate any moisture remaining in the wood after cleaning, and the spores you weren't able to remove will go dormant.
Seal the wood with a single coat of clear finish to lock the dormant spores into the grain and isolate them from moisture in the atmosphere. It's a good idea to scuff the old finish first by sanding by hand with 320-grit sandpaper. Wear a respirator to avoid breathing any spores that you release when doing this.
Control humidity in the room in which you're storing or using the oak furniture. You can increase ventilation by opening a window or using a fan, or you can run a dehumidifier. Cover your freshly cleaned and sealed oak furniture with plastic sheeting if you have to store it in the basement or some other place where humidity is difficult to regulate.