If you don't see any screws, this could be a moderately big job. Unfortunately, a number of these fans are not easily removable. Often the fan is held by brackets that are spot welded to the box -- so they are not removable from the box -- and the box itself is attached to unseen brackets that are nailed or screwed to the joists. If that is the case, and since there is no access from above, the only way to remove the unit is to cut away enough sheetrock until you can get at the brackets. This means you'll have an oversize hole and have to do a sheetrock repair when you replace the fan with a new one. Even many new fans are still manufactured in this way. You might be able to use a power jig saw with a bi-metal blade to cut out the fan and hack the box into pieces, but that would be difficult and you would have to be careful to not cut into the electric supply wires and/or damage any vent work.

What are the interior dimensions of the box? Do you know the brand?

I'm not clear on whether you want to remove the whole shebang or just the motor--not that I can even see a motor.

Is there attic space above or is there another story above? If attic, go up there & look see.

Immensely helpful for us if you want to remove the entire thing would be to vacuum out all of the dust bunnies & post new photos.

I replaced a cheapo bathfan that was installed in the first floor powder room. The housing was attached to the joists with just screws, accessible after the vent cover was removed. There is too much debris collected on your fan for me to be able to see if yours is attached similarly.

How forceful were you in pulling the fan out when depressing the clip? I believe that is the release but it will take a bit of convincing with 6 decades of rust holding it in. I'd slide a piece of metal under the clip to scrape out accumulated corrosion.

I agree with HU...43 that if that is indeed a spring clip (at the top of the second photo) it would take a lot of force...maybe prying with a flat edge screwdriver to move it out of the way. But even if you can get the old fan out of the housing, the likelihood of finding a replacement for it after all these years is, unfortunately, remote.

Thanks for the thoughts. In response to the requests for more info:

1. The box is 7 inches by 9 inches. No idea of brand. Likely 60 years old. Still works, though noisy.
2. I have no access from above.
3. The 2 clips on the side are not what's holding the motor in. I can easily release both sides with screwdrivers but the fan motor still feels solidly in place.
4. I have cleaned out the dust and am attaching additional photos.
   Thanks again for any ideas that don't involve creating a bigger hole in the ceiling!
   
   

Based on the info you just provided, I continue to be of the opinion that the fan is not something you can disconnect from the box and replace. Since it's a lot smaller than typical joist spacing, the box probably has an inaccessible bracket you can't see that is on one of the sides and is screwed to a joist, or there is some other kind of inaccessible bracket across the top that's adjustable to match the joist spacing. In either case, it's going to be nearly impossible to get the assembly out without cutting into the ceiling. However, many newer fans are enclosed in boxes that are larger than 7X9, so you have some leeway even if you damage the edges of the ceiling opening.

The only other thing I can suggest is that you cut through the sheetrock with a multi-tool all around the edge of the box. It looks like the four sides of the box each have a half inch flange that rests against the sheetrock; once you cut away that small amount of sheetrock, it's possible that you could exert brute force upwards and break the fan box free of the bracket.

Good luck. Beyond this, I'm out of ideas ;-)

After you muscle it out like Kudzu suggests, this fan might work for you:

https://www.build.com/air-king-bfql50/s1324341?uid=3156858

You don't need attic access to install it and it is a bit larger than the current fan.