Can I run Algolia on-premise (on my own servers)?

No, Algolia is only a SaaS offering. 

That being said, when choosing our ENTERPRISE plan, we setup your account on a dedicated cluster instead of the common shared ones. This ensures that no other customer will be hosted on the cluster hosting your service, and that your cluster is optimized for your specific needs.

In all of our plans, you can also select the location of your cluster in one of our worldwide distributed datacenters.