Is it possible to run Google App Engine Development Server on my own server?

YES, with a lot of missing features (parallel queues, cron jobs, mail, XMPP,..), some hidden security issues, poor performance and stability, it is technically possible.

As you probably guessed, it's a bad idea.

Take for example the HTTP server; using the development server you would put in Production an undocumented BaseHTTPServer, quite impossible to configure and with probably some hidden security flaws ready to be exploited.

As @Drew well said, there are better choices out there to run you Google App Engine code in a Production ready environment that is not GAE.