Is there any way for me to run my own instance of a Google Wave server?
Yes, Google has said they intend to release an open source Wave platform and intend for people to run their own server and to roll their own clients.
Check out the video at Youtube they go into quite a bit of detail about it.
The Wave Protocol web site will have all of the information and specifications needed to build your own Wave server. Whether you decide to make it available to talk to others is entirely up to you.