An outdoor steam room is entirely possible and considerably more work than an outdoor sauna, for three reasons that all come back to water: the generator cannot freeze, the enclosure has to be waterproofed against weather and steam at once, and there has to be somewhere for the condensate and the drain to go.
Freezing is the first problem
A steam generator holds water, and water in an unheated outbuilding in a cold state will freeze and split it. The answers are a frost free plant space, a drain-down routine every autumn, or a heated cabinet. None is free and all need deciding before the build, because retrofitting a plant space is the expensive kind of change.
Two waterproofing problems at once
The enclosure must keep steam in and weather out, and the two want different details. You need a proper roof and drainage on the outside, a tanked and sloped enclosure on the inside, and a way for the cavity between them to dry. This is why most outdoor steam installations are masonry or purpose built pods rather than converted cabins.
Where the services come from
A water feed, a drain and a dedicated circuit all have to reach a building in the yard. That is a trench, and often a permit. An outdoor sauna needs one cable, or none at all if it burns wood, which is the whole reason outdoor saunas are common and outdoor steam rooms are not.
What most people should do instead
Put the steam indoors, where the services and the frost protection already exist, and put a sauna outdoors, where it needs neither. If the outdoor experience is the point, a wood fired cabin with water on the stones gives you humid heat in a back yard for a fraction of the work.
Questions people ask about outdoor steam room
Can a steam generator live outside?
Only in a frost free space, or with a reliable seasonal drain-down. Water left in a generator through a freeze will split it.
Why are outdoor steam rooms rare?
They need water, drainage and a circuit run to the yard plus two kinds of waterproofing. An outdoor sauna needs one cable, or none if it burns wood.
Is there a simpler way to get humid heat outdoors?
Yes: a traditional cabin with a stone load, run with water on the stones. It is not a steam room but it is humid heat in a back yard.