Masking external links and make them only available for members?

Almost anything is possible with custom coding. Where do these links occur? Post content? Output by template code? Output can be conditionally altered based on the return value of is_user_logged_in(). How it is altered depends on where the links are coming from.

On post content.
Some links: abc.com
Some links: example.com
I’d like to cover them with mysite.com
and make it only available to members.

You can alter content just before it is output through the “the_content” filter. Your filter callback function can do a regexp (regular expression) search and replace for all links in content with something else, perhaps a link to a page encouraging visitors to become members. Of course only do this if is_current_user_logged_in() returns false.

The PHP regexp search/replace function is preg_replace(). For example:
$content = preg_replace('!(href=")(https?://.*)(")!', '$1mysite.com/recruit/$3', $content );
This only matches href attributes that use double quotes. You can prevent links from being replaced by ensuring the href attribute uses single quotes. For example:
<a href='https://not-replaced.com'>
The example regexp also only replaces http and https protocol links. Links with protocols like mailto: and tel: will not be replaced. Naturally the regexp can be fine tuned to fit your specific needs.

The sort of code that does this can be placed in functions.php of a child theme or some site specific plugin that contains custom code for the site.