Technically, you're talking about two different things here.
Quarantine tanks are used to temporarily house newly purchased, presumably healthy fish just in case they're not healthy. The goal is to avoid infecting your main tank. Most folks quarantine them for three weeks. These can be set up and taken down as needed.
Hospital tanks are for sick fish in your main tank. Obviously, you'll want to use these to isolate the sick fish from your healthy fish, but also to isolate your healthy fish, inverts, and plants from the medication(s) that you'll have to administer. Ideally, these should be left up and running permanently, but real life often does not allow for that. Some folks run a sponge filter or second HOB filter in their main tank and move it over to the hospital tank to "instant cycle" it. (Keep in mind that a cycled filter doesn't have the bioload capacity of a mature cycled tank, so you'll only be able to house a few sick fish at a time.)
You can use the same tank for both tasks. Just be sure to sterilize everything in between uses.
I have a cheap 10 gallon "kit" aquarium that I use as a quarantine tank. I cycle by seeding with filter media from my old tank and go through a short cycle by adding ammonium chloride solution. This takes a week and a half or so. I like to add some rocks, a ceramic cave, and fake plants. They make the fish feel more comfortable and they're easy to sterilize. They also provide extra surface area for BB to grow. Since the tank has a relatively low-volume HOB filter, I like to use an airstone to help speed up the cycling process.