How long does an opened glass jar of anchovies last when refrigerated? It was preserved in olive oil and salt.

I trust you will want to use them before the end of the world comes. I have had some stored in the fridge for year with no problem, but they are usually eaten way before then.
You must make sure that all the anchovy is covered with oil. Any little bit sticking up out of the oil will start to grow a mold...eventually. Push the remaining anchovy back into the oil and add more oil if necessary (a good idea anyway). Do not worry if the oil solidifies - it will liquefy in ten minutes when you pull the jar out of the fridge.

Indefinitely. Of course, after a while the anchovies will start to break down and become mushy, but you can still use those in pasta sauces and the like. If you see any mold or discoloration or it smells funky (well...funkier than normal), I would toss it.