Can You Transfer Files To A New Mac OS From A Old One that's in Recovery Mode?

One day my Mac OS Intel-Based stoped working.

I went to disk utility there is a problem with the disk I can't fix..... (it did say it was complete but I looked into . it one of the process was failed)

Moreover,I don't want to erase, damage any files until I get it transferred over to my new Mac.

Is there any possible way you can do this?

My Full Story.

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One day I booted up my Mac OS-Intel

the question mark file Icon came up, Below that was link-If your Mac starts up to a question mark - Apple Support

I went there to follow the steps.

However if you looked deeper there were issues with the main disk,I did not realize that at the time.

I went to download or reinstall Mac OS when I got to the third tab "Mac OS Mojave select disk you want to re- . instal Mac OS"

However, No disks appeared in that tab it was just blank.

I got many recommendations to erase the disk however, I don't want to do that because important files are the computer.

If you have any recommendations, or ways I can fix this please reply.

Thank You! :)