Feedly is not available as a self-hosted solution but there are plenty of alternatives for power users and business that want to host a solution on-premise. The best Self-Hosted alternative is Tiny Tiny RSS, which is both free and Open Source. If that doesn't suit you, our users have ranked more than 100 alternatives to Feedly and many of them are available as a self-hosted solution so hopefully you can find a suitable replacement. Other interesting Self-Hosted alternatives to Feedly are FreshRSS (Free, Open Source), CommaFeed (Free, Open Source), Miniflux (Free, Open Source) and selfoss (Free, Open Source).
FreshRSS is a free, self-hosted RSS aggregator. It is quite lightweight, fast (it can manage +100k articles without complaining) and powerful (e.g. shortcuts, filters, responsive design, multi-views, multi-themes, multi-users, statistics, provides a Google Reader API, etc.
Google Reader inspired self-hosted RSS reader, based on JAX-RS, Wicket and AngularJS. Deploy on your own server (using TomEE, a lightweight JavaEE6 container based on Tomcat) or even in the cloud for free on OpenShift.
Looking for a simple and open feed reader? Free, open source, easy to use, Sismics Reader is what you need to keep your feeds under control. Central Access all your RSS and Atom feeds in one central easy readable place.