Try:
git clone git://git.moodle.org/moodle.git mystuff
cd mystuff
(If I understood your question correctly)
You can also quite safely do the following:
git clone git://git.moodle.org/moodle.git
mv moodle mystuff
Git stores all its settings in [checked-out-repository]/.git - so you can move / copy the main folder as much as you like without breaking anything.