Ok, Fedora adheres, thankfully, to the FHS a lot and, thus, changes where things might go. In this case, they don't use the git user; they use gitolite3 as a user. And, in a nutshell, here's how to configure it:

The first part of the HowTo goes on the Server:

# Become root
su -

# Install gitolite3
yum -y install gitolite3

# Become the gitolite3 user
su - gitolite3

# Copy the public part of the key you're going to use to manage this gitolite instance (from your workstation).
# Obviously, this has to be a single line and no spaces. This is, normally, the source of all problems: badly formated keys
cat < < 'EOF' > renich.pub
ssh-rsa AAAAB3NzaC1yc2EAAAADAQABAAAEAQC8LgmK... My Workstation Key
'EOF'

# set it up
gitolite setup -pk renich.pub

Now, from your Workstation:

git clone gitolite3@host:gitolite-admin

That is dead simple! Just do it and have your own gitolite3 instance!

Now, read the manuals in order to get literate about it's management and configuration:

ACK

Thanks, John Hall, for the correction ;)

Thanks, John, for the correction on the clone command as well.