Note: I actually bought a low cost unmanaged 8 port switch before posting this question as I suspected it was the hardware required. I have now found out how to set the network up. It is fairly simple.
All other PC's should be configured all auto with DHCP. The OS X computer needs slightly more configuring.
I suggest testing the below steps with a single computer connected without a switch first, as this removes one extra layer of required debugging.
Search google for something like "share Mac OS X wifi internet via ethernet lan connection" to will find more info on some of the following steps.
Test that the OS X computer can receive an internet connection via WiFi.
Turn firewall off. Note this step is important and caused me hours of work to figure out why it didn't work. You can re-enable firewall later, see below for details.
Connect OS X pc to other PC via single (tested, working) ethernet connection.
Go to System Settings on OS X -> Sharing -> Internet Sharing
Set: Share connection from Wifi via Ethernet, and click checkbox on left to enable. Confirm enable in warning message.
You should be able to access external sites on the other Linux / windows / etc computer through the shared network connection.
You can now re-enable firewall on OS X with the following settings:
Go to System Settings -> Security and Privacy -> Firewall
Turn on firewall, options: Allow incoming (no checkbox on block incoming), automatically allow signed software (will populate the listbox above when signed software is automatically added), do not enable stealth mode. (Last option had me stuck for ages.)
To create a basic "local network", add an unmanaged network switch. This last step may seem trivial, and indeed it is, if you know that you just have to plug it in and there is no other configuration required.
I was lost in a rabbit hole for some time searching for things like "how do I set up a DHCP server on Mac OS X" - I thought I had to setup my Mac to emulate a Router, DHCP and DNS server, but this isn't required as the sharing option does it all automatically.
I added this as a later addition. Advantages include routing traffic from several wireless connections via a single wireless connection, which is likely to be more efficient. I believe the approximate explanation is a reduction in wireless collisions. If using a gigabit switch, traffic between those computers connected via the gigabit Ethernet will be considerably faster and lower latency. On the other hand, latency between computers routed downstream of the OS X computer and the cloud will be slightly increased.
Possible problems with this method
I believe the sharing options creates a separate subnet on the ethernet shared side. If I am correct, this means it would be impossible to SSH, for example, from a computer in my room to another computer in the house. I believe this is because I would have to transfer from one subnet to another through the Mac OS X computer. In which case my assumption is that I would have to setup port-forwarding through the Mac? I intend to test this later this evening and report back on this.