Why I might get this high noise on Sony a7 III with Zeiss 24/70 f4 lens? So I've tried to make a short New Year video and noticed, that I got noise at almost every level of ISO over 3000. And it is the kind of noise, that is impossible to work with. 

Am I getting something wrong, or it has to be this way?
For example, this short video shows how high the noise is in both light and dark. 

The params are: 1/200 f5.0 ISO:4000 120fps/60m 1920/1080

Can I reduce the noise, or is this something I can definitely expect with 4000 ISO and such lens? Do I have to light everything in a way, that I won't need to go over 500 ISO? 

I'll be glad to any suggestion and answer.
You need to define what is acceptable to you.

Record a well illuminated scene and increase the ISO while decreasing the aperture and compare what noise level on your specific camera is acceptable for you, and you will know what are the limitations of your equipment.

In the case of the example video, the noise is "the same" because both moments, the dark zone and the white are filmed in the same settings, therefore we can assume that is the overall noise on your camera on that ISO setting.

There are some programs than can denoise video https://duckduckgo.com/?q=video+denoiser&amp;t=h_&amp;ia=web one that has a good reputation is Neat Video.