While having your own Apache host or similar is technically feasible, it is not without difficulties (as Grimaud as already commented), especially for someone of limited expertise in this area.
How attached are you to using Wallabag? If you'll excuse this left-field suggestion...
Have you looked at note-taking apps? A lot of these have "web clipper" features or plug-ins that let you capture web pages or portions of pages.
Check out OneNote, it has lots of possibilities and free account has 5GB of space. I don't like it, but that doesn't mean you won't.
Check out Google Keep, and Evernote. Obidian gets good reviews and definitely has a clipper available, and it has a free option - but I'm not sure of its limitations.
I was recently looking at these and they frustrated me because they were all based on the same premise as Wallabag: offer what looks like free software, but make practical use of it contingent on subscribing to a cloud-based storage service.
I already have 15GB of mostly unused space on each of a couple of Google accounts, and these days I have 1TB of space available on OneDrive (as part of a Microsoft 365 subscription)! So why can't I make effective use of all this space to which I already manage passwords and so on?
Which leads me to: I have ended up trying out
Joplin. It lets me configure where the notebooks are stored (or I can have multiple profiles, each using a different storage), and these include DropBox and OneDrive, amongst others. Plus the more security conscious amongst us can configure client-side encryption so you don't even have to trust the storage provider.
Joplin has a "Web Clipper" plug-in available for common browsers. I can't tell you that much about it other than that it works. I tried it out but it's not what I use Joplin for so it doesn't get much use.
I'm not all that impressed by Joplin's Android app, but it works well enough for my purposes. Their Windows app is good, if not great. And they have versions of macOS, iOS, and Linux, but I haven't used those.
I'd also suggest looking at the
Vivaldi browser, it has a built-in note-taking feature with the ability to clip content from web pages ... but while Vivaldi does have a (secure) sync feature, I'm not sure how comprehensively it covers notes; somewhere in the help I saw mention of not synchronising note attachments, so this may limit its usefulness for your purposes.