If you want as little finish smell as possible—for example, on cabinets in a house or office that is occupied, or on the inside of a box or chest—there are two finishes to choose between: water-based finish and shellac.
Other finishes, including oils, varnishes and lacquers will require days and maybe longer for all the residual odor to dissipate.
There are two major differences between water-based finish and shellac. Water-based finish is more scratch resistant and imparts no coloring to the wood. Shellac will scratch easier and add a warm yellow-orange tone to the wood.
Both dry rapidly, so you can apply multiple coats in a day. But you have to work fast if you are brushing, especially on complex surfaces such as the chair pictured.