Edit 1 based on the current version of the question "Where can heat be vented from an air conditioner that isn't outside, in the attic, or into the ground?"
Answer: It is possible to put the heat into something like a large water tank. The problem is that you have to take the heat out of the water tank eventually. This is the heat battery concept. In a practical sense, when your available resources is that you have a room you can't even fit a regular air conditioner to, you're just going to be hot.
You can also try to use an evaporative cooler. These are sometimes called swamp coolers. Here's the issue: These only work in very dry environments. In most of the USA (assumed from your use of degrees F), using one of these will just turn your room into a sauna.
New question 2: How does a ceiling fan fit? A ceiling fan, being an electrical device, will add some amount of heat to the room. This is probably insignificant compared to the environment and other loads. What a ceiling fan does is makes it feel cooler in the room because moving air cools you down, at least as long as you can sweat. There are edge conditions in the southwestern US where blowing very hot air across your skin actually heats you up, but in most places, blowing a fan at you makes you feel cooler.
Let's examine what we're trying to do.
An air conditioner is a device that pumps heat from cold to hot, getting by the whole heat-flows-only-from-from-hot-to-cold thing. The cool thing about it (no pun intended) is that you can move more heat than the energy you put in to the device.
Now, let's discuss what the heat is we're moving. Heat is typically divided into latent (humidity) and sensible (temperature) heat. Your heat in your room comes from several sources:
- environment sensible heat (it's hot outside!)
- environment latent heat (humid air infiltration)
- civilization-induced sensible heat. Pretty much every amp of electricity that goes into your house gets turned into heat. The biggest in America tends to be the refrigerator. But every single thing that uses electricity turns it into heat.
- biological-induced sensible and latent heat. Humans, dogs, and cats are hotter than a cooled space, and thus produce heat. Their breath and sweat also produce latent heat.
So, what can we do with this heat we are removing? Practically, almost nothing. The expense to turn that low-temperature heat into work that you or someone else could use is more than it's worth. Other answers note the idea of heat batteries. If you happen to live in one of the places where it is hot during the day and cold at night you could store the heat and let it out at night. The thing is, it's probably easier to just let the hot daytime heat up that battery anyway (desert communities have done this through building materials for millennia). Adding complexity to electrical/mechanical systems costs more for minor paybacks.
We could do things for fun, like Peltier devices that make enough electricity to maybe charge a phone, or Stirling engines for the fun of it, but these aren't scalable solutions.
So, in summary, just enjoy the cool and worry about other things.