How do you store peeled ginger?

It freezes beautifully.

I dry it well, then wrap in a paper towel. It will easily keep a week in a very cold fridge. Also, immerse in a jar of sherry, as suggested by Diana. The flavored ginger keeps forever and goes well in stir-fries.

You may want to consider slicing it thin and candying it; then you can carry it with you wherever you go, enjoy it anytime, and never have straggling amounts in the fridge. Personally, I love a slice of candied ginger in my cup of black coffee -- it gives it a slight sweetness and ginger flavor (surprisingly good with coffee) and then I have the rehydrated slice of ginger at the end (not quite the same as eating the worm at the bottom of the Tequila, but that doesn't seem quite appropriate first thing in the morning either!). ;)

I submerge mine in sherry or sweet wine and keep it in the fridge.

For a few days in a ziploc in the fridge.
For a few weeks, submerged in vodka(makes a fun martini) in the fridge or wrapped and frozen