As already mentioned by
@rucxy, drop some ducting in the trench along with a draw string. Use wide swept bends as it makes drawing cable much easier. After all, ducting is a trivial cost in comparison to paying someone to dig a long trench...
Once the ducting is buried in the ground, then many many choices are available as to what to actually pull through (if anything). Who knows, if you go the wifi Path, such may be enough initially but interference and poor wifi performance may cause you to rethink your solution. Even wifi security issues through lack of device firmware update may give you pause to rethink your solution. At that point, the small extra cost of that ducting already in the ground will make absolute sense...
But in terms of connectivity between the house and shed, I‘d consider drawing some optic fibre through the ducting. I’d then use
these devices (
more here) at each end to convert between fibre and copper. Fibre length can be upto 500m with such “SC“ devices (which in your case is more than enough for your 35m + what’s needed for the building internal parts of the run).
Fibre cable isn’t expensive, around $3/m for 2 core multimode fibre, but the terminations are seen as complex and unfamiliar by many so they shy away from it. Luckily, due to the number of NBN contractors out their, I’m sure there are many that can advise and then terminate the cables for you at reasonable price.
I like the philosophy of fibre as it’s unimpeded by proximity to power cables since light isn’t impacted by electromagnetic fields. It’s also future proof in that changing the devices at either end of the cable will allow you to get increased throughput with what’s already in the ground. As an example, when 10G copper to fibre converters are cheap.y available, they can be used to swap out the existing devices giving a speed boost from 1G to 10G ethernet bridge between the house and shed.
If I was doing a shed and wanted internet, I’d explore the fibre path... but if I ended up going copper, I’d run Cat6A as it has much better shielding than even Cat6 and can support 10G Ethernet over 100m runs (Cat6 allows 10G over short 35m-50m lengths based on noise environment). I wouldn’t draw a Cat5 cable (as there is hardly any price difference between Cat5, 5e and 6 while Cat6a is $0.40/m more costly at around $180/m; google should find prices but note I’ve listed indoor cable while outdoor UV stabalised cable is much more expensive but not needed in under ground ducting)