How can I hang a ceiling fan on a cathedral ceiling without wiring or an electrial box?
After writing this long answer, here's a YouTube video.



I'm not a professional, but I had to do a ton of research for a similar situation at my home, and here's some information that I hope will be helpful.

First, it would be easier to break your question into parts, for clarity.  I may be misunderstanding your question though.

Q1: I don't have electrical in my ceiling, how can I get power there?
Answers:

First there may actually be wiring up there, an electrician (or you) could figure it out by trying to get access to attic or space around the perimeter of the cathedral ceiling.  In our house, from the attic we see Romex cable entering the cathedral ceiling in between the rafters.  So I would do some research there first, or ask a Pro too (they would know where to look faster).

If there is no power there already, it should be possible to add it to an existing circuit from a nearby wall.  An electrician would cut a hole in the ceiling and then cut a hole in the wall and use a fishtape to bring the wiring up to the ceiling.  I'm not a pro, so there's a lot of detail here that is missing, but trying to give you a general idea (there are a ton of resources online to find out how to do that).

Worst case scenario you could surface mount the wires like another person mentioned, but unless it is a rustic house or you have a very limited budget, I wouldn't go with that (unless you are ok with it as a short term solution).  I would think an electrician / GC could get wiring to this location for $300-400 and might install the fan for that price as well.

Q2: how do I attach the fan to the ceiling?
Ceiling fans need considerable support.  If the center of the room or the location where you want to mount it doesn't have a joist/rafter that you can mount the fan to, you will need to get one of these mounting brackets:



Westinghouse-15-5-cu-in-Retrofit-Ceiling-Fan-Saf-T-Brace (search that on google)

I used something like this (not sure the brand), and it works well (it is very stable/solid), but it definitely involves cutting into the ceiling, however all the work can be done from the one carefully cut hole it needs anyway. Also, depending on how your ceiling is designed, there may be no clearance space (ours has insulation between the exposed ceiling and the roof deck).

If you have a joist/rafter that you can attach the fan to, then you can get an normal mounting bracket (your fan might come with one already).

This is my first answer on this site, hope there's some helpful info!

My references:
familyhandyman.com/electrical/how-to-install-ceiling-fans

They have a good video on installing the fan brace too, but stack exchange won't let me post 2 links