Random thoughts . . .
As mentioned a cord is 128 square feet (or another way of thinking is wood stacked 4 feet high by 4 feet wide by 8 feet long) . . . to build a shed that would hold a single cord of wood you would want something that would fit those dimensions . . . just remember that depending on the construction type (i.e. using a traditional wood stud wall like I did for example vs. large corner posts) you may want to build the shed a dite larger since you will lose some of that space due to the construction (i.e. due to walls I lose 8 inches or so on the sides and 4 inches on the back wall.)
I'm also a believer of going larger if you can afford to do so and if you have the space . . . I find it is always a good feeling to look into a shed and have wood left over by the end of the burning season vs. being left with nothing or having to dig out wood from under the snow. Any wood left over from one year can always be used in the next year . . . plus a larger woodshed means you may use it for other purposes (i.e. I added a snowmobile lean to shed on to my woodshed for my brother in law to store his sleds.)
If you truly only want to build a shed for one cord of wood and are looking at two rows of wood you have to decide how tall you want to stack the wood, how long you want your shed and how deep your wood will be . . . I generally figure my wood to be about 2 feet long -- namely since I use 20-22 inch wood and like most guys tend to round up when talking about wood

-- I suspect a shed 4 feet wide by 6 feet tall and 6 feet long would work for you . . . but again . . . if you're building and using standard 8 foot lumber you might as well go 4 feet wide, 8 feet tall and 8 feet long . . . it will give you more headroom, more storage for wood and fewer cuts.
Building a woodshed is a relatively easy project . . . it's a simple box or rectangle with a half sloped roof on top . . . of course there is a little more to it than that . . . such as making sure your corners are good and the whole thing is tied together . . . but honestly, it's a relatively simple construction project . . . and the true beauty is that since it is "only" a woodshed it can be as fancy or as simple as you want . . . if you do a search you'll find woodsheds cobbled together out of pallets or scrap wood and other fancier ones that are stained or painted with fancier roof lines and even electricity . . . the truth of the matter is that simple or fancy . . . they all do the same job -- keep the wood high and dry and allow you to avoid using tarps or other covers.
Foundation . . . as Muncybob mentioned some folks dig holes and use "sauna" tubes to create footings below the frost lines . . . others just use blocking above the ground . . . some of this depends on your terrain and soil which may limit how much the frost will move the shed . . . and sometimes it just is a matter of folks not caring if the shed is not perfectly level ten or twenty years down the road.
Being able to go to Lowes or Home Depot and get a shed has a certain appeal . . . but I'm not sure if the floors would be up to the weight of the wood . . . wood is heavy . . . which is why many folks go with concrete, gravel or rock pads . . . or use pallets for the floor . . . you can use wood . . . but in my opinion it should be plenty beefy to carry the weight of the wood.
Best advice . . . do a search on this site for woodsheds . . . and if you see a shed that you like and it seems like the right size or you can scale it down . . . ask the builder if they could give you more details on how they built it since there truly are a lot of ways to build a woodshed.