Mirrored servers in data centers nationwide -- how?

I am mixed here. What google does is standard anycast.

http://en.wikipedia.org/wiki/Anycast

Basically you have your own internet addresses, in your own AS (a term a sysadmin should know) then assign all clusters the same IP ranges (all data centers). DNS will round robin, but you do not use this to direct to a specific addresss.... you use bgp anycast announcements for that. What you need to is access to the internet gbp level, which requires you own AS. The trick is that if a DNS says "address x" then your DNS anycast setup routes all requests to the closest data center.

That said, it is not standard knowledge for a nommal sysadmin to know about that. As you can see on the wrong answers further up who seem to not be even aware of Anycast, which is pretty much a standard approach. BGP / internet backbone routing protocols is not something 99% of the network admins have any need to know. In that, the question was unfair or for a senior position.

Also, what daemons run on these machines 24/7 to keep all those mirrored database disks synchronized?

None. Discs are not synced. To my knowlege google machines run in clusters of X machines each (forming one functional unit) and culusters dont get updated ever. They get regularly reinstalled with a new and updated image. Between installs, no sync happens.

Was this interview for a sysadmin position? If so, I think the question is slightly unfair. Granted I don't know the full breadth or context of the question, and you should definitely know how components such as DNS and load balancing work in general, but I don't think you could be expected to know how Google (for example) manages to make it's services globally diverse and highly available.

Mirrored servers in data centers nationwide? That could mean anything.

What needs to be mirrored? A web site? A Windows domain infrastructure? An FTP service? Etc., etc.

You may want to look into load-balancing and CDNs.

There is no one specific daemon that I'm aware of. There are however a number of ways of doing this.

Edit: And to answer your comment, DNS is not the ticket here. Its all about CDNs(content distribution networks) and load-balancers, redirecting to a number of available, mirrored hosts. These hosts are mirrored through any number of technologies, including(not likely in Google's case) rsync and things of that nature.