The first thing to say is that few animals do well when they are kept sopping wet. This includes frogs, which go into water to mate or as an escape from land predators, but which are not really aquatic. Tadpoles are, but not grown-up frogs, which can drown
The thing is, all animals have ways of avoiding drying out. They may shelter under rocks, or in leaf litter for example, but when you keep them captive, you may limit their flexibility. So to keep them moist but not wet, the trick is to keep them in a humidity jar.
Raw plaster, which you can buy in packets at the hardware store as "Plaster of Paris" can absorb water and a layer of plaster at the bottom can help keep a jar from drying out. Here is what you need to set this up:
On that last item, I generally use coloured cardboard as a backdrop for photography, and here, it serves a double purpose.
Add some water to the clean jar. The amount is not all that important, but I suggest 1.5 to 2 cm. Some people suggest drilling a few holes in the plaster, but I have never bothered doing this.
Then add some plaster to the jar. Notice the spillage in the picture on the left? It always happens, so it's a good idea to sweep it up before it goes on the floor. Use a dustpan or a vacuum cleaner and never use a damp cloth!! (If you do, you will make matters worse. Trust me.
You want a layer of plaster 8 to 12 mm deep, and there needs to be water on top of the plaster. Don't worry about that, because the plaster will still set under water. I always bang the jar on the bench a few times to get any air bubbles out, and to get the plaster surface level.
After 20 to 30 minutes, take the lid off, pour out the remaining water, wipe the inside of the jar with damp paper, and your humidity jar is ready to put some animals into it.
If you want to keep tiny web-making spiders, choose a suitable twig a and poke its thick end into the plaster while it is still wet.
The sky's the limit: even leeches can live there...but then what do you feed them on?
You can see one solution to leech feeding, on the left ...
In case you are wondering, it was involuntary feeding, but my wife is a biologist and called me for a photo opportunity before we removed it.