On one Minecraft server, I had some moderator powers. I intended to ban someone for 3 days, but it turned out to be 3 seconds. (Vanilla Minecraft doesn't give this option; if someone is banned, it has to manually be undone.)
Then there are the really short blocks on Wikipedia. If the admin blocks the wrong person by accident, it can be under a minute, but there was also a time (thought it was mentioned on either WP:STOCKS or WP:LAME, but it's not) where someone was blocked for 30 seconds intentionally.