This isn't possible because a git repository contains a whole entire projects contents and changes. If you took one directory, then each commit would potentially have a different subset of changes, making the signature of each commit different (you could probably write a script that does this for you, but its not how git is meant to work).
Instead, you would have to download the directory and initialize a new git repository with it. Currently, GitHub only lets you download an entire branch's (or commit's) repository as a ZIP archive. So something like this:
# Download and extract archive
wget https://github.com/bob/thisrepository/archive/master.zip
unzip master.zip
# Copy subdirectory out of repo
cp -r thisrepository-master/path/to/subdirectory /path/to/new/project
# Create a new git repo
cd /path/to/new/project
git init
git add .
git commit -m "intial commit"
Bonus:
Check out this answer for information on git-archive
and how you can use that to extract the sub-directory directly from GitHub, rather than downloading the whole repo and extracting (you'll still need to initialize a new repository based off of the files).