Apple - Using my own server instead of iCloud

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.

The simplest way to accomplish equivalent functionality is to set up the following items. ^{This isn't meant to be a full iCloud dopplegänger but instead to serve equivalent high level functions of iOS backup and iOS file sharing endpoint.}

1. Disable iCloud backups and instead back up to iTunes via WiFi.
2. (optional) Set up a VPN profile on iOS to have the device be able to VPN in to the local WiFi while on the road.
3. Set up WebDav server - optionally expose it to the internet with either a static IP address and/or one of the dynamic DNS services.
4. (bonus) set up a cloud or single point backup so that the server is backed up offsite. Look at CrashPlan or Amazon's various storage options (glacier or S3 are written up in several places on the internet with "How To" and their new cloud drive offering might also work well)

This allows DropBox like file sharing over WebDav - your own personal cloud that is optionally backed up outside your house periodically (but not a true cloud where your iPhone could hit multiple storage tiers at once). You could then decide if backups while on the road are needed and set up/test the VPN access to iTunes for networked backups.