can I copy the code and host my form on my own server?

You can indeed get the full source code of your form:

https://www.jotform.com/help/104-How-to-get-the-Full-Source-Code-of-your-Form

This is not fully self hosted since all back end processing would still be done on our end.

Good News,

Jotform Enterprise is released and it supports hosting your data on your private dedicated server, "Single SignOn" and "White-labelling". You can get further info on https://www.jotform.com/enterprise/