I think you could make it work. You might talk to your neighbors first though, lol. Unless you have a big coyote/dog problem, I wouldn't bother with fencing the gap, but it wouldn't hurt anything. Your future goats would certainly enjoy some straw or wood shavings for bedding.. make it deeper during cold times of the year. Do a google search for "goat hay feeder" and you will get lots of ideas for simple hay feeders. Unless you live someplace that doesn't have bugs, your biggest problem I would guess will be flies in the summer. You'll definitely have to keep the bedding fresh in that type of space to keep the flies from going crazy.
If you're wanting to do milk goats, you could just feed them their grain on the stanchion. If not dairy and just "pet goats" you could just use a feeder like this:
http://www.enasco.com/prod/images/products/85/AC105835l.jpg
And again, just some type of round tub or large bucket (depending on how often you want to change the water) clipped to the wall somehow (so it doesn't get knocked over and they are without water for hours while you're gone) will work just fine for water.
Oh, and the last thing... any plants, shrubs, grass, trees, etc you have in your yard will be fair game to your goats. Make sure you do not have any poisonous to them plants back there either.
They 100% will eat your trees, your rose bushes, anything that is palatable to them that they can reach.