Using my own server instead of iCloud

I am currently paying like a dollar a month for 20 GB iCloud storage, but I am just now building a new computer and am hoping to install an Amahi server on it so that I can have my own 'Dropbox' type of cloud access wherever I go.

Am I able to use a service like Amahi (or anything else) to redirect my iCloud backup from apple's servers to my own? Instead of paying $12 a year for 20 GB of storage, I could be paying no subscription for 3+ TB of space.

In a nutshell: No, you can't.

Apple uses proprietary protocols for iCloud access (both data and backup) so you can't just switch backends by redirecting some DNS entries. And as far as I know, nobody has reverse engineered the protocols yet (at least not the full set required to implement an alternative server-side solution).

I found OwnCloud, which does at least appear to give the option to host your own cloud on your own server, that is to sync files across devices: desktop, Apple, Android etc.

It's open source and free, with iOS/Android apps around $1 USD / $0 USD respectively.

It has been thoroughly tested on many different architectures.