I participate regularly on another Beta site, which is probably around the same size as yours. At any rate, it's pretty small! There are a couple of moderators there who close and delete questions unilaterally when they feel it necessary. So I checked what the rules are, to share the information here. Here's what I found:
What special privileges do diamond moderators have?
Their votes are binding. Any place that requires a consensus of multiple users — closing, reopening, deleting, undeleting, reviewing, marking as spam, etc — a single moderator's vote will reach the threshold and take effect immediately.
I'm a bit confused because I don't see these three names listed in the comment by @Narusan-sedated:
The problem is that there are only a limited amount of active high-rep users: You, L.B., Lucky, Carey, Kate, Mark, Doctor Who. I will be gone for the next 5 weeks or so (enjoying my vacation rather than voting to close questions here) so out of 5 out of 6 people who could vote to close will have to vote to close to get a control of the situation.
And I do see diamonds next to the three moderator names at https://health.stackexchange.com/users?tab=moderators.