This post is "picking the brains of smarter than me" more than anything. It's a simple question, but I can't seem to find a consistent answer to it. I have heard sooo many different suggestions and so many different "right ways" of doing this, that every way I try, gives different results.

How do you properly setup an antenna to measure it using a VNA (when you don't have access to an anechoic room

Please watch my video and comment below.
Video of my trying different techniques

 

First of all, you have no ground plane with that antenna. Even a handheld provides some "ground" for the shield of the antenna. Your setup has no ground at all, just the cable running back to the VNA. That causes lots of RF currents in the cable itself.
You could try an aluminum plate with a hole in it, or even aluminum foil as a ground plane. tightly clamped to the ground of the antenna. If all the antenna has is the SMA connector, you have to build a mounting jig to mount the antenna on, one that has a ground plane.
Moving the battery pack (with -Ve being ground) changed things, you are placing a ground near the antenna, which couples to that ground.
Same for getting your hand close to the VNA, you are coupling yourself (possible ground) to the VNA, which is acting as a ground for the antenna.
Grabbing the ground of the cable far away is still coupling the "ground" of the antenna (which has no ground plane) to your body.

All these things will change the resonance and SWR of the antenna. Your test setup is just plain wrong, the antenna is not mounted properly with a proper ground plane within a few mm of the antenna connector (as in a handheld). Your shield of the cable is probably radiating (since there is no ground), and touching the outside shield changed things as well...

 

This post is "picking the brains of smarter than me" more than anything

Hello there
you should ahere to safety precautions. Do not touch a VNA.

 

There are two types of VNA in the markets today. One is SA1201 having one port but up to 2,7 GHz and another is two port device named NanoVNA for up to 900 MHz. It exits in two versions, smallish screen and much better size screen. Generally the antenna demands no anechoic chamber. Just plug it and read the readings. However, if the antenna is far off from 50 Ohm impedance there may become visible the myriades of false resonances in spectrogram, what happens on cabling corners, schtekkers etc. So, to make a correct readings antenna must be near to 50 Ohm. By the way, there are possible to measure any capacitor, inductor etc passive elements, and You shall be surprized that 99% of all capacitors and resistors are more the coils not a n element You considered it to be. Only after obtaining VNA I realized importance of blocking capacitors for 3 Eur per piece while beforehand I was buying for 0,3 cents per piece. Fantastic device and how much I made intercomparizon with far expensiver apparatusses, it shows rather exact figures accurately.

 

Prerequis:VNA Analytical Tools, Signals and Propagation . time domain vs. frequency domain. In S-parameter theory, we denote Smn = bm an , where ak = 0 ∀ k= n.
I would I would destroy that video you made.