Create a local bare repo, you can (put in Dropbox folder) use like remote.

create-bare-git-repo.sh

# Create a new bare repo.
cd origin-parh
mkdir repo.git
cd repo.git
git --bare init

# Make an already existing repo bare.
cd origin-parh
mkdir repo
cd repo
git init
cd ..
mv repo/.git repo.git
rm -rf repo
cd repo.git
git config --bool core.bare true

# Working with a bare repo.
cd working-path
git clone origin-path/repo.git
echo 'Changes' > new-file.txt
git commit -a -m "Commit"
git push origin master

# (Re)setting remote
cd working-path/repo
git remote remove origin
git remote add origin origin-path/repo.git