How Do You Refinish Marble?

I think you would have to get a professional to do this, as specialised equipment is needed to refinish marble and polish it.
Although marble is considered a soft stone, and is more porous than other stones, (hence it gets damaged and stained by acids that react with it_, it is also brittle. It can easily break or fracture, if knocked or dropped.
The polishing discs that are required to put a gloss and shine on it, are big specialised machines, you need to take it to them. There should be a marble finisher in your city, either for countertops or gravestones, look in the yellow pages.