The easy way to get around this* (*most of the time)
I came across a site the other day that put up an overlay when it detected I was running ad-blocking software. My usual tactic (depending on if I'm making a one-off visit to a site or see myself coming back again) is to either use Chrome DevTools to delete the offending HTML or write a Stylish rule to permanently hide it.
This site, however, was different. It detected any attempts to make changes to the DOM, either manually or with Stylish, and immediately rendered a completely different error page.
In the end, I simply edited the site permissions (using the icon in Chrome's address bar) to not allow JavaScript on the website. Problem solved.
Obviously, this won't work well for sites where JavaScript is required for certain interactions.