As you're aware, Google won't do the job without an internet connection. Even the offline version of Google Docs only allows local changes, not synced ones.
Perhaps surprisingly, Microsoft's Excel does allow simultaneous editing by a number of people on a local network. It isn't real time like Google, but it does update everytime users save. There are simple instructions on this blog post, the relevant part being:
The workbook needs to be in a network share and it must have a
write privilege to all your team members.
- Open your workbook.
- Navigate to Review on the toolbar
- Click on the Share Workbook icon
- Place a check on the check box that says Allow more than one user at the same time.
- Click on OK
- Now, save your workbook on a network share
While far from perfect, if everyone is encouraged to save regularly it might be an alternative for your situation.