Hi
Have built several structures over the years and the runaway best choice has been extruded aluminum. A shadehouse is going on 20 years and still almost looks new.
At the time it was more expensive than wood but I doubt that now.
Woods that don't rot such as cedar and redwood are incredibly expensive at least here.
With the aluminum there's no worry about strength or decay .
I keep birds in the large shadehouse so know about rats lol. I use an aluminum frame with 1/4 inch mesh . Rats , so far (knock on wood) have been unable to get inside, Have had them actually gnaw through wood. gary

With the latest in chemical treatments that eliminated poisons and as an added protection for both the wood and anything that might come in contact with it; a good coat of penetrating oil stain sealer, I really believe the concern over treated wood and soils or what grows in them is really unwarranted.

However, if you construct beds with treated wood, lining them with heavy plastic should also be good insurance, although I also believe the plastic encourages rot, as it traps moisture. I line my treated timber walled beds with weed blocker fabric, which keeps the soil from sifting through any joints and allows drainage and haven't been been downed by a home grown strawberry yet.

Nevertheless, if you construct a space that will be walled and want to keep out rats, you will also need to prevent them from digging under to get to your garden and for that, nothing will beat 1/2 hardware cloth laid just under the foundation timbers all around the perimiter to about 18 - 24 inches out from it, with the outer permiter edge of the wire firmly held down to prevent any rats from finding a place to crawl under it.

They always try to dig in right at the base of the structure and we have yet to have any predator or pest smart enough to think to back off to find the outer perimiter and tunnel under it to breach the walls, like furry sappers.

That line of ground defence has been defeating the desires of our local weasels, mice, foxes and coyotes to enter our poultry yard for over 40 years, and Wiley Coyote is a lot smarter than any rat.