To be clear, absolutely zero of this is required for a hobbyist. This is all if you want to do professional level, repeatable, objective measurements/reviews.
Does it actually require that much gear to get
measurements that are accurate enough that no
human can perceive the difference?
Yeah and no.
For speakers, yes, you need some of this sophisticated gear if you want to do objective (as opposed to purely subjective) reviews.
The Klippel NFS is an automated rig that measures a speaker's output from many different angles and uses some nifty math to subtract the room's influence, giving results more reliable than a physical anechoic chamber. That rig costs something like $100K; I forget.
Getting a 360 degree picture of a speaker's sound output is pretty important. The majority of sound we hear indoors is reflected. You are, essentially, hearing the sum of what your speaker is doing at every single angle all at once so if we're going to characterize a speaker's performance in an objective way we need something like that.
Of course, you don't need need a $100K rig. You can measure a speaker in your suburban back yard (or better, an open field) from a few different angles and get a pretty decent approximation of all of that. I've done it quite a few times! All you need is a $60 calibrated mic, a protracter, a tape measure, and some free software. And the willingness to annoy your neighbors with loud tone sweeps. Usually takes me a few hours to measure a speaker; most of the time spent is just dragging things around so I can do 20 minutes of measurements from 5 or 6 different angles.
However, if you're looking to do this with large numbers of speakers and have lots of money like Amir does, the Klippel certainly starts to look attractive.
For measuring DACs and amps, no, I guess you don't need a professionally calibrated AP analyzer. I think you can spend a couple hundred bucks and get pretty close. Maybe less.
For measuring headphones, yeah, I think you need something like the GRAS headphone measurement rigs that are $6K-$13K. Or maybe the $299 one from MiniDSP is just as good. I don't know!