You can easily go to a sauna with temperatures exceeding 110° C. Hot as heck, but even kids do it.
The starting temperature of World Sauna Championships was 110° C, with water/steam added every 30 seconds. The competition ended up killing a person, tho.
I'm surprised that sweat doesn't start boiling at that temperature, or that the users don't get severely burned since I assume everything in the sauna is going to be around that temperature --- even 50C is enough to burn skin:
My sauna last night was 84 C, 183 F. I personally don't like much hotter. The higher the temperature, the more you have to modulate the steam that you produce by throwing water on the stove. At a lower temperature I like lots of steam, but at higher temperatures a lot of steam at once can be too much. Steam is controlled by the users on a moment-by-moment basis while temperature is controlled more slowly by the stove settings (which of course you can change).
The wood in the sauna does get hot: that's why you either sit on a piece of cloth (more popular these days) or put cool water on your seat to cool it down so your butt doesn't burn. However, wood in saunas is generally carefully selected as some varieties feel cooler than others (and you never want to use a wood that will produce sap drips for your benches! ouch!!). There are almost no metal components to touch in a sauna because those hurt -- some people use copper buckets for water but then you're filling it with cool water intermittently.
Take off metal necklaces and earrings before saunaing, in general. Don't know about piercings (someone else can say). You can cook/burn yourself in a sauna but that is usually related to drug and alcohol use (as in the famous World Sauna Championship death: https://www.theguardian.com/world/2010/aug/08/sauna-champion...)
Sweat evaporates in the dry air, and that cools your skin. A bit. The wooden benches tend to get too hot to touch, which is one of the reasons why you sit on a towel. Glasses, watches and other objects are left outside. (I'm not sure how big of an issue piercings are, actually. If they get too hot, take them out.)
Still air is a good insulator, it doesn't transfer heat to your skin very quickly. And skin has good cooling via blood circulation and evaporating sweat. These cooling mechanisms also transfer more heat as the temperature difference grows.
> You don't actually get in the water at 80°C right?
No, you get in the water below 10°C. The 80° (or 90°C, as I often see claimed on Finnish saunas in Netherland) is the air. Extremely dry air, where your sweat evaporates almost immediately. Until someone pours on the löyly.
