As I was explicitly asked to, I'll turn my comment into an answer – but for the entire question incl. details from OPs comments:
Using the latest LOS (and AFAIK Graphene, too, soon), you have Seedvault built-in – which is to Google Backup what microG is to Google Services. And yes, the Cloud part you can self-host. You might expect what comes Next, cloud?
If a custom ROM is not an option, there's always my little Adebar using ADB via command line (Linux, Mac, Cygwin). While it's not performing the backup itself, it creates scripts to be used with adb to create separate backup archives for each app supporting to be backed up (which is the same group of apps the Google Backup would cover). It also creates a nice documentation for your device, see here for some examples.
Now, why should a custom ROM not be an option? You've pointed that out in your comments, so let me counter that:
- microG gives you most of what Google Services cover, minus the tracking. If your device is officially supported by LineageOS, you can pick your ROM from LineageOS for MicroG and have it included right from the start (if not, see below¹). By default, all ties to Google are disabled – but you're free to enable Cloud Messaging, SafetyNet etc. from its settings. It even offers you network location using a backend of your choice – I eg have the OpenCellID database for my areas on-device, so I have network location even without network connection :D
- you miss your 5375 apps? I don't see the problem here. There are plenty of apps to access different "stores". F-Droid is constantly growing (more than 3.300 apps already). And you still can access the hell of that walled garden you so urgently want to get into (no offense meant ;) using e.g. Aurora Store – yes, funnily you get that app at F-Droid. So yes, all those app stores I can recommend (ahem, that's only 1, make your guess) and even those I wouldn't recommend at all should be usable from LineageOS.
Going this path, you can even add "privacy respecting" to the list of attributes. Which is what your question really is all about, right?
¹ here's "below": if your ROM doesn't ship with microG, or is not even a custom ROM, you can use NanoDroid to get microG installed easily. NanoDroid comes as an "all-round carefree package", getting rid of proprietary apps and replacing them by good open source alternatives, each of your choice (so you choose what shall be gone and what come in). Find instructions there, please; it requires Magisk and some work, though.