Replacing clutch?
 

Is this possible to do by jacking the car up? How high off the ground will the car need to be, in order to do this? And does it take any specialty mazda tools?

Basically looking to see if I can do this myself, since the clutch is only 300$, but labor costs can exceed 1000$.

I would assume that it's possible to do and that the car isn't significantly odder than a normal RWD car. You might want to find a copy of the factory service manual to see if anything varies from what I've written.
Remove shift knob
Remove the clutch line
Possibly remove the clutch release cylinder
Remove the starter
Remove the exhaust
Remove the Driveshaft
Support the transmission with a jack. (you might need to unbolt the PPF, I have never messed with a car with one)
Unbolt the transmission mounts
Unbolt the engine to tranny bolts
Remove a few bolts that attach the clutch to the flywheel, pull clutch
It's a good idea to have the flywheel resurfaced. Remove the nut with an impact gun. Alternatively, lock the flywheel using a chain attached to a bolt where the clutch attaches, and somewhere else for the other end (possibly the tranny bolt holes), use a long breaker bar or attach a pipe to a 2" bar.
Reinstall flywheel and torque to spec.
Use a clutch centering tool to install new clutch - make sure it's the right way.
The rest is the reverse of the removal.
Bleed clutch line.
I'm probably forgetting some steps, but those are most of them. :)

It's it technically difficult to replace a clutch? No. You do need to jack the car significantly up in the air. You need axle and/or wheel stands. Is it a physically ackward job? Yes. I would say that you need to be reasonably mechanically inclined to want to attempt this. Alternatively, you could always try it and should you run into trouble then tow it somewhere to finish the job. :)

I wrote a "Clutch/Flywheel Replacement Experience" thread a few weeks ago when I did my clutch & flywheel upgrade:
https://www.rx8club.com/series-i-trouble-shooting-95/clutch-flywheel-replacement-experience-170426/

This should give you an insight on what to expect.