Oil has to be re applied every so often and you need to prep and clean the furniture every time you do and could be as often as once every 6 months.
Personally I've used fence paint on mine and it's been fine now for several years.
Fences don't require painting every 6 months and neither do sheds.
So use shed or fence paint. I bought the largest pot for peanuts out of B&M and it's black paint. It comes in various colours.
If you want to keep the natural look then your looking at spending like £100 on whatever they use on log cabins. Tuin has a good guide for that and they sell stuff too I believe.
Don't buy specialist paint fence or she'd paint is what you need and will work best.