I've got a Chuwi Minibook (8inch screen) for that kind of usage and happily runs Linux on it nowadays (older versions of Ubuntu used to periodically hang which I believe was an issue with a driver for the disk).
There was a brief side line on netbooks where you could swap in a daylight-readable display of the sort used in the OLPCs, too, but I never had the confluence of money, time, and availability to make that happen.
I don't really know if there's any way I can use it for anything nowadays. Windows is out of the question - it originally shipped with Windows 7 Starter, which it was already struggling with, to be honest.
I tried installing Linux on it, and it turned out that it requires a proprietary graphics driver, support for which was dropped from the kernel at some point...
Nowadays this form factor is gone, it got injured with how Microsoft tackled the market they were initially losing, and got the final blow when tablets with detachable keyboards became widely adopted.
That is what normies are using all over the world in coffee shops, if not a regular laptop.
Likewise, now a Samsung tablet has taken over the role I used my netbook for.
I added 4gb of ram to it and an SSD and it's ok for light loads.
It uses some AMD processor that has no cache or something ridiculous. C-50 Or C-52?
It also has a sticker for "HD Internet".
Worked great until I updated Linux to discover that I could no longer install the stupid proprietary driver for the graphics, so lost hardware accelerated graphics. Was never able to downgrade successfully.
I've been idly looking for something modern to replace it but seems like there's nothing like that on the market now.
All she wanted was an actually portable Windows device, and Asus delivered.
Celeron was as hot as iPhone for a moment then! There was a real hype around this device category ("Netbook" was the German marketing term)
This machine was key to helping me learn Backtrack Linux (Kali's predecessor) and introduced me to Fluxbox.
Great concept though. I wish the format would have stuck around. I'd like a premium version in that same size.
When things go wrong, the Eee is the backup to step in and fix things.