Would you nowadays write a new web service in Perl?

Perl was my primary language for eight years and I can't genuinely think of a good example. Even in situations where there are fantastic packages for certain tasks, like with Mojolicious for WebSockets, you can do the same in Node, Go, Ruby, Python, or pretty much anything else now, so it ultimately comes down to preferring the language in most cases.

No, not in a thousand years, so I'm curious what situation might be tempting you

User facing? No. Backend batch job services? Absolutely. Taint checking and strict would be globally mandatory. Every service would be aware of all other services and degrade gracefully.

I wouldn't, because I have a different language of choice, but you should use whichever stack you can best maintain and make work for the use case. If that's perl for you, go for it!