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Staining your pallet wood is the best way to treat it for outdoor use. In order to give your project the touch of a perfect finish, you are going to pass through the wood pallets staining process.
I would use Thompson's Water Seal.
This might help a bit: https://www.1001pallets.com/staining-your-pallet-wood-tips-for-beginners/
seal it how? insulate it? weather proof it? wrap it inside & outside with canvas painters tarps(keeps out bugs and wind) then place strawbales all around the outhouse for added insulation ; for even more insulation in frigid temps mud the outside of strawbales
I would use a wood sealer that is meant for outdoors
Why not paint it with some good semi gloss enamel after priming of course. If it was the same color as the house it would blend in more and not look so homemade.
Use deck paint
I would put vinyl siding on it to seal it .and maybe if you need to add sheet rock to the walls or t1.11 to the walls so there waterproof it’s whT they poor in shower walls
Helmsman Spar Urethane. Used on wood boats
Hi! Use a wood sealer meant for exterior use. There are some excellent ones available. Good luck and stay safe!
Marine varnish if you want to keep the current look of the wood. It will change slightly but that's probably your best method.
Uee a good water sealant
Use deck stain for outdoor use.