Installing MacOS on external hard drive and booting to recover files

If your laptop is running macOS 10.12.6+, then you can boot into Internet Recovery Mode (Command + Option + R) to access the online macOS installer. Sometimes you can use regular Recovery Mode (Command + R), but sometimes it will only access a really old version of macOS which may have trouble installing and may not be sufficient to read the SSD (may need to upgrade to a later version to read the SSD).

Here is the link for creating a macOS USB installer mentioned by @Ronasara:

https://support.apple.com/en-us/HT201372

Another option if you have another Mac is to try using Target Disk Mode or you can remove the SSD and install it into an external USB OWC Pro Envoy Enclosure. This may be your only option if the laptop has hardware issues.

I assume you have an internet connection. If so, then go to the Apple web site. It has links to download various operating systems. (I'm sorry, I don't have the link.) Then go ahead and download whichever one you've selected. It may take up to an hour to download to your Applications folder. Your desktop should also have a window in which you start the install. At the beginning, it will ask you to select where to do the installation and you then select your attached external drive. The installer will then do the installation on that drive which can also take a long while. It will restart several times.

Before doing the download, make sure that your external drive is properly formatted by the Disk Utility app for APFS.

Then you can use the Migration Assistant app to transfer your data from the old HD to the external drive. Be sure that you have a wired connection such as a USB or ethernet cable and do not do it via wireless as that does not transfer well.

You still should look at having another external drive just for backing up everything.

I hope this answers your questions and helps you. Good luck.

By all means give that try. You never know what works until you try.