Yes, the low disk space warning actually prevents from stopping downloads and causes to run out of the disk space!!!
Turns out, I have run out of the disk space and this warning popped up:
"Newsbin Paused - Low Disk space on Download folder Drive"
So I paused all the downloads ("Paused" on red background) and pressed OK to get rid of the warning box.
Unfortunately, when pressing OK, downloads resume (the red box goes away), they do not remain paused.
And then I pause downloads immediately again, but still after some time the same warning box pops up again.
It pops up even though downloads are paused at the time (I can understand why, it is not the issue here).
But the only way to get rid of this box (which remains on top of all the windows) is to either press OK or to press X in its corner.
Both actions cause downloads to resume again - WHICH IS INSANE - and so on.
The disk space eventually goes down to zero literally, thanks to this warning box.
This is clearly a bug - if downloads are already paused (manually by me) they should remain paused after the warning box is closed.
If the downloads were halted temporarily by this warning, then after closing it they may resume, I guess.
The only work-around, I guess, is to make more free disk space while this annoying box stays on the screen and only then close it.