I have a Mac that was using macOS High Sierra

Please refer to Use macOS Recovery on an Intel-based Mac - Apple Support. In Step 2 under Start up your Intel-based Mac in macOS Recovery, availability of the first option "... to reinstall the latest macOS that was installed on your system" appears to have been removed by Apple.

Choose one of the two remaining startup options: "upgrade to the latest version of macOS that’s compatible with your Mac" or "reinstall the version of macOS that came with your Mac or the closest version that’s still available" respectively.

You ought to be able to install macOS Catalina on that Mac, which — should — still be available on Apple's Recovery servers.

As for High Sierra, it appears to be gone.

Time Machine backups of it, assuming you have been maintaining them, are another option.

I can't help with the recovery server error message, but be aware of the following.

A 2013 iMac can be updated to the latest version of macOS 10.15 Catalina. See the following from Identify your iMac model - Apple Support

iMac (27-inch, Late 2013)

Model Identifier: iMac14,2

Part Number: ME086xx/A, ME088xx/A

Newest compatible operating system: macOS Catalina

To upgrade, see: How to download and install macOS - Apple Support

Be aware that if you upgrade to Catalina 10.15 or later you will also need to update any non-Apple 32-bit apps. See: https://www.stclairsoft.com/Go64/

or

 https://www.macworld.com/article/232735/how-to-check-if-mac-software-is-32-or-64-bit.html

If you can't or don't want to upgrade your macOS, for security reasons you should consider using a 3rd party internet browser that supports their current version with your older macOS. One example is Chrome if you have macOS 10.13 or higher. See: https://www.google.com/chrome/