Andrew,
If you can pick up wifi from just outside the shed, perhaps get something like a wifi extender/access point and set up an external antenna outside the shed to repeat the network or convert it then back to Ethernet inside the shed.
I have a backhaul set up for work over 500m that works in a similar way with two TP Link CPE510's, but you don't need something that is good for 200km for this.
Joel
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