A couple of days ago I updated Apple Mail but after the update its hung up. Nothing seems to be working. After I launch it, I am not able to send or receive any emails. I am asked to reset it by some professional but before resetting it I want to backup all my data. I am not relying on Time Machine as well because it starts backing up everything on the Mac. Looking for a decent solution to backup apple mail but how can I do it if it's hung up like this.
You can go directly to the folder where Mail saves its data through the Finder and duplicate it to store a copy elsewhere. Inside the Finder, menu Go > Go to Folder and then paste this path:
It is that V5 folder you'd want to back up before reseting Mail. There is also some software solutions like mailbackupx but I never used it so I can't say what it's worth.
Try rebooting your system, that might help.
Also, there are multiple apple mail issues discussed here, might be helpful, have a look: