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Witnesses operate the servers. It‘s explained here on the top of the page: https://peakd.com/me/witnesses

Well thanks, I checked and it said my witnesses would expire in 24 hours so I voted again. Since they are servers, does that mean I would no longer be able to use HIVE anymore and would be locked out had I not voted?

No. You could use Hive without a problem. And you can vote for new witnesses anytime.