Will SQL Server go offline if it loses network connectivity to SAN where master and msdb system databases reside? I have a setup where 3 servers combined into Availability Group
All 3 servers have local directly attached SSD drives, and user database files are hosted on these drives

But the system databases (master and msdb) of each server in AG, are hosted on a SAN device that is accessed over the network

Did not move those to local SSD drives yet
Questions:
In a hypothetical situation where network connection between any of the servers and the SAN device is lost (bad cable, bad NIC, some temporary network glitch etc.),

Will SQL Server service on that server go offline or stop working properly immediately ?

Or it continues to work for some time if master and msdb were cached in RAM before network went down ?
You can't depend on the system databases being cached in memory, more likely they won't be because of their less frequent access relative to the user databases.
I think you'll end up in a quasi-functioning state, where your user databases will still be accessible but certain features of the server instance that rely on master and msdb will throw some weird errors, depending on what else your server is doing. The service for your SQL Server instance, should continue to remain online (&quot;started&quot; state). For example, if you have any Scheduled Agent Jobs, I would bet my money (but couldn't say for sure without testing) they'd encounter errors (either silently or apparently) when trying to run since most of their meta-data is stored in the msdb database.
If such was to occur, you'd be best to restore access to those system databases as soon as possible to guarantee 100% reliability in all features and functions.