We don't know where you are located but fences don't need staining 2 x per year in my climate. Once every 5 - 10 perhaps. And you can get organic and or non- toxic wood preservatives.

Old pressure treated wood could leach heavy metals into the soil in a minimal amount but it has been off the market in the US since 2003. New pressure treated wood is far less of an issue as it doesn't include any heavy metals in the processing, however its use in a vegetable garden is not accepted under certified organic standards, which have little to do with the vast majority of home gardeners.

If you have ongoing concerns, use a naturally rot-resistant wood, like redwood or cedar. And agree that you can address staining or sealing with any number of non-toxic products. Actually, both of those woods look great and last just as long if left unstained or sealed.

or a cement board , hardiplank will make a long lasting fence and holds paint, stain for a LONG time. It does not leach either.

If you are in the country or your area's large, there are hedges. Evergreen for year-round privacy and they provide protection and habitat. Or just not stain the wood and replace it when it gets worn out. I think cedar ones last at least 10 years if ignored. Nontoxic paint works too and only is put on every few years.

I like these vinyl fences. Beautiful possibilities. No maintenance, but lots of class and you can choose your own style.

http://fencesbycountryestate.com/