There are two different treatments you can have of this outdoor structure, and each has its own set of problems. Local codes will probably determine which you must actually use.
If you treat the outdoor structure as a separate electrical system, it must have its own ground and panel. In that case, your LV Ethernet cable can potentially support the voltage difference between the two electrical system's grounds, which in the event of an electrical storm can become a huge voltage. So there's a potential life safety issue if you touch that cable, or even something downstream of it. Your entire Ethernet network could end up spreading a lot of charge. You need input protectors on both ends of the cable, MOVs properly grounded. Check your local codes, because this might simply be something you're not allowed to do at all.
If you treat the outdoor structure as an extension of the main structure's electrical system, you bring a ground wire to it. In this case, the "ground" of the structure's electrical system can become different than the potential of the actual soil around it, leading to shock hazards. The farther away the structure, the more potential you might expect to have, and the bigger the problem. I'm not sure how you deal with this. The entire scenario is probably not really compliant with electrical codes for any nontrivial structure (this is more how you'd deal with a lamp post or something like that). The good news is that, in theory, both ends of your Ethernet cable are connected to devices that are all part of the same electrical system, and as long as there's a fat enough feed to the outdoor structure and it's wired right, you should not have excessive potential differences except in cases like a direct lightning hit.
The moral of this story is that you need an electrician who understands your local electrical codes to figure this out for you. There are local code/legal issues at work, and there are life safety issues at work. That's a nice way of saying that you, or a member of your family, can be electrocuted, or suffer an otherwise nasty shock, if this is done wrong. So please, do this right.
As with most electrical safety things, done wrong will work fine.... until someone gets hurt.