Best Way Of Storing Ancient Copper Coins?

If you wipe them with mineral oil they will become darker. Usually the visibility of copper coins is less when they are darker. Mineral oil is a bad idea.

If you live in a very dry region (like Arizona) you need not worry about "bronze disease" or PVC problems so much. If you live on the coast of Florida with high humidity you have to avoid the sort of soft plastic flips that auction firms often use. Hard flips, which protect against PVC problems from soft flips, can scratch coins, so auction firms may use them to avoid having viewers accidentally scratch the coin when they pull it out and return it to the flip. Many collectors use hard flips ("PVC free") and replace any soft flips they get.

I have seen collections of many serious ancient-coin collectors, none of whom use 2x2 mylars. Most want to be able to get their coins out of flips and handle them and look at the surface without having to view it through highly reflective mylar.

Mylar 2x2's allow the full identification of the coin to be attached to it.
That way allows for the moving of the coin from one pocket to any other, and the identification travels with it.
Very convenient, when you re-oragnize your collection from time to time.