Now with a fully vetted files app and new file sync provider API on macOS and iOS it’s much easier to bring whichever file storage solution you want. Automated device backup is still challenging, but ensuring your main file storage hit other accounts should let you ride the free iCloud tier for backups - especially if you can turn off categories you don’t need backed up to iCloud due to cost considerations.
In 2015 and the following couple years, it was much harder to avoid iCloud and then, the simplest way to accomplish equivalent functionality was to set up several things.
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.
- Disable iCloud backups and instead back up to iTunes via WiFi.
- (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.
- Set up WebDav server - optionally expose it to the internet with either a static IP address and/or one of the dynamic DNS services.
- (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.