This is probably a dumb one, but here goes.
I've got a tiny house with one bathroom, and it only has a shower. I love taking hot baths, used to do it 2-3 times a week, more often when I'm sick, and I miss that. I have a pretty big shed in my back yard, think one of those pre-fab things... it's got some tools and stuff but it's mostly empty space. It's pretty close to a hosebib and an exterior electrical outlet on the side of my house, so I've got cold water and extension cord access.
Is it even reasonable to consider ways to set up a tub in the shed? Maybe freestanding? maybe something I build from fiberglass? I'm pretty confident I can drain the water out into my backyard with no issues (big yard, neighbors won't see / care), but is there anything obviously impossible or really stupid about this idea? Is it possible to use an electric water heater or should I figure out something with propane / butane? Anyone tried anything like this before?
The concept is doable for sure, but I'd reconsider the draining issue. The volume of a hot tub is quite something, and if you're using any kind of soap or other chemicals, you've got a potential environmental challenge on hand. Even if you only use biodegradable stuff, that doesn't mean that plants will rejoice being smothered with soapy water.
For the heater you'll need something with a decent wattage to get it up to temp in short enough time, so make sure you're not overloading the circuit. Those outlets in some shed are often not the best.
Propane/butane would work as well, but in that case you'll need to take care of proper ventilation as well. Or just leave the door open.
You don't put soap in a hot tub. You'd end up with a backyard full of foam.
You primarily put bromine in. I wouldn't dump it on your vegetable garden, but its not an environmental challenge.
It's safe to water most plants with if you give it a day open to let the sanitizer evaporate out: https://texashottubco.com/can-i-use-my-spa-to-water-my-garden/