For some reason, I'm unable to log into eBay, Hotmail, Paypal, or Facebook. My digital banking is fine though.
Tried this from two machines, my iMac and this G4, both with no luck.
Those sites various apps are all working on my iPhone, using the same wifi network connection as the iMac.

I'm stumped.

With seemingly random and intractable connection issues like that I sometimes resort to a complete shut down. Everything gets turned off, and even unplugged, even the ADSL model and any routers you have.

Then I turn it all back on, one bit at a time, starting with the ADSL and network bits. Rebooting the Mac last, letting it fully start up and settle (this takes some minutes on this old MacBook Pro).

If there's still no joy, I'd look into the Mac's network control panel and try renewing the DHCP lease (Network>Advanced>TCP/IP).

If that doesn't work, I'll have to join you in the flummoxed corner for a bit.

I've done the shutting down thing. Router, Airport and computers.
I'll have a poke at the network panel.

I've also flushed all the cookies, and reset Safari.

Just ran the Keychain first aid, and it's all verifying as ok.

What is the same about the two machines?

We know the following so far:
- They are different architectures
- The iMac is running Mountain Lion
- The G4 is running Leopard or earlier
- The G4 doesn't belong to you
- Sites (e.g. eBay, Hotmail, Paypal, or Facebook) can't be logged in to on either machine.
- Browsers used: Safari, Firefox, Chrome
- An iPhone has no problems

Are the profiles synced at all? I.e. are any prefs from the iMac also on the G4?
Are they running the same versions of Firefox/Chrome/Safari?
Is the iMac and G4 on the same network?

So far I'd say it was the network, but seeing as the iPhone doesn't have problems, I'm somewhat at a loss...

RIght - so - changing the MTU to 1432, after enabling some remote management features on the router, appears to have done the trick.