You will end up downloading the entire history, so I don’t see much benefit in it, but you can checkout specific parts using a “sparse” checkout.

The steps to do a sparse clone are as follows:

mkdir <repo>
cd <repo>
git init
git remote add -f origin <url>

This creates an empty repository with your remote, and fetches all objects but doesn’t check them out. Then do:

git config core.sparseCheckout true

Now you need to define which files/folders you want to actually check out. This is done by listing them in .git/info/sparse-checkout, eg:

echo "some/dir/" >> .git/info/sparse-checkout
echo "another/sub/tree" >> .git/info/sparse-checkout

Last but not least, update your empty repo with the state from the remote:

git pull origin master

You might be better off using a shallow clone too. Instead of the git pull command given earlier, try:

git pull --depth=1 origin master