Time Machine to NAS, over internet

Solution 1:

Synology does have this walk-through for doing what you want to do.

I will mention that (quite some time ago) I used a Synology NAS as a time-machine target disk, and I did get it to work, but found it to be highly error-prone. Not what you want in a backup. Also slow. I'm using a drive with a direct USB3 connection now and am much, much happier with that.

Solution 2:

This is not a substantial answer, maybe treat it as complementary to the first answer.

I should expect Time Machine over AFP to be relatively reliable with a LAN, relatively unreliable or impossible when limited to a WAN (wide area network).

In the Time Machine Network Interface Specification (TMNIS): Time Machine Server Requirements. I guess that whilst a server may meet those requirements, an aspect of the WAN will defeat a requirement.

The size of bands, within a sparse bundle disk image, may be ill-suited to AFP over a WAN. And so on.

Solution 3:

I wrote a guide to configuring Time Machine to back up to a cloud server which is almost relevant. And it looks like the DiskStation supports the same IPSec server setup that I'm recommending, so you might be able to crib some parts from my guide to make it work.

Time Machine over the internet works for me, even over a 1Mbps connection connection. Though you have to leave your Mac running for a bit longer to make sure each backup completes, so number of completed backups is going to be proportional to that bandwidth.