If you're already relying on Google, how about creating a Google hang-out and joining the user to that?

If you want to control chat yourself, you can spin up an existing chat server (such as ejabberd) on a cloud or VPS host like linode or Amazon ECC or interserver.net or whatever, and use XMPP for the chat. This will scale to many thousands of simultaneous chatters; it will not scale to many tens of thousands of simultaneous chatters.

Building your own is possible (and necessary once you get into the hundreds-of-thousands of users category) and is a challenging task. I ended up using Erlang to do it a while back; that worked out OK but I'm not sure we really saved a lot compared to a plain C/C++ solution in the end.