Networking G5 to Mac Pro

I don't think wireless in your case has the range, and you say it's too far for Ethernet cable.
Do you have power in this shed sourced from your house, if so the maybe these power line adapters might work for you.
I'd pick them up from a store that has a good return policy, so you can try them and return them should they not work.
http://www.cnet.com/topics/networking/best-networking-devices/power-line-adapters/

 

I use TP Link 600mb power line adapter through out the house and can't fault them.

 

Assuming your G5 is in range (which I highly doubt it is) you would turn on Internet Sharing on the MP and tell it to use the wireless card.

On the G5 you'd connect to the signal the MP is broadcasting. If the G5 can see the MP through the walls of the shed and your house to connect to the MP then you'd be good.

If not, then there is the powerline option as discussed above. Another option is to use a wireless bridge. Essentially, you slave a wireless router to your main network. That router you'd place in the shed and you can either hook the G5 up to it via ethernet cable or connect to it wirelessly.

I have two wireless routers in my home network that are acting as bridges to my wired router. It's a fairly simple process and if the router you have has a wireless bridge mode it's even easier.

 

You should have 2 or maybe 3 green lights depending on the model of power lines you have. To test I would manually set the ip address on both the g5 and the Mac pro