That message means the game is for some reason not connecting to BZRNet. There haven't been any outages recorded, which means the problem is likely on your end. First port of call is to make sure you don't have a firewall installed that's preventing the game from accessing the internet.

You need to use gog galaxy.

Hi, maybe someone know, cam I play BattleZone 98 Redux with my friend (not versus, but coop store mode)?

Why are you making a reply in a random unrelated thread?

There is MPI mode but it relies on specifically-made maps that only exist as maps made by the community and that are largely only available on Steam Workshop as GOG lacks any equivalent service. Some stuff might be found on ModDB.
There are ways to obtain Steam Workshop content for a GOG copy of the game. Someone made a tool for this.

Direct Download:
https://github.com/Nielk1/Battlezone-Redux-Mod-Manager/releases/download/v0.4.0.1/BZRModManager-v0.4.0.1.zip

Main GIT Page:
https://github.com/Nielk1/Battlezone-Redux-Mod-Manager/tree/v0.4.0.1

Hello, maybe someone will help to understand - can I play BattleZone 98 Redux Story Mode with my friend (not versus, but coop story mode)?

No

Unfortunately there is no such option (. I know it sad, because its so cool game and it would be great option to play it in coop together (^_^)!