To replicate:
Clicking the X to close the modal does nothing. Hitting ESC does nothing. Reloading the page seems to be the only way to close the modal.
This site has a few crazy plugins, but the problem persists in safe mode.
I have just given this a go, and it works here on Meta?
Cannot repro… maybe you should look at those crazy plugins?
Feel free to PM me the hostname and I’ll be happy to have a look.
OK. I moved this to support. This guy insists that he hates custom plugins but also insists that his site does things that require plugins. 
I will crank up the staging site and start removing stuff.
It was the guest gate plugin. Thanks, @RGJ!
I’ve looked at a couple of customer sites recently which had some truly egregious examples of plugin hoarding - mores than certain Wordpress sites. The need to rebuild when installing a plugin had convinced them to install practically everything they could find on the off-chance they would one day need them, and unlike WP they didn’t appreciate that even disabled plugins can have an impact on stability and performance.
Interesting. A two-container installation an a dashboard that would let you graphically add/remove plugins could be a good solution for those people.