fridge is no problem..stay's good for a while. also u can put it in a small container and put some oil on top, good part of that is that your oil is going to taste like Anchovies to..use that for dressings...
To be safe, it's good to put it in a container with a tightly capped. Anchovy can last long with a maximum of two weeks in a refrigerator or even without refrigerator.
Tightly capped or not, I would NEVER store opened anchovies or anchovy paste without refrigeration. That's an easy invitation to violent food poisoning.