Sounds like spammers. There are a few things you can do to discourage spammers. I took a look in your control panel, and it looks like you've already taken a lot of these measures--was that in response to the signups?
You should be able to re-enable your site, but keep these options enabled:
1.Require CAPTCHA for registration, here: https://ducktalk.net/cp/registrationSettings
That should discourage a bunch of them.
2. You can moderate registrations (you'd have to allow new registrations again, though, which you have disabled on the Registrations Settings page). Moderating registrations would just allow you to see the registrations as they come in, and you can choose to approve them, or just see them when they come in, so you can ban them if they seem to be a spammer.
3. You can also make your site private (which I see you've also enabled), but I'm assuming you'd prefer not to do that. The other options above should help you considerably.
Were the signups posting spam? You can create recipes to moderate some obvious spammy content for the spammers that do slip through the above.
Let me know if you'd like to set up a moderation recipe, and I can help with that.