No, web pages can access local storage.
(Can't say I'm terribly up to date on the different ways to store data as a website... but if there's a way to persist multiple gb I would be very surprised)
> You mean file:// urls? If you have access to do that...
Modern browsers provide JavaScript with way more power than just file urls: https://www.w3.org/TR/FileAPI/
There are also drag and drop APIs...
> why not just install a real executable as well.
That depends on who your audience is. It's certainly easier to tell some users "go hit this web page and click ok when it asks to open your file". And there are some web apps which I find preferable to downloaded apps. For instance, I really like draw.io, and it works seamlessly on Windows, MacOS, and Linux.
> but if there's a way to persist multiple gb I would be very surprised
The only limitations I'm aware of are tied to this nonsense about 32 bit integers :-)
As for "what is the point if not doing computation on "real" data sets"... letting you do computation on reasonably sized data sets. Consider that the size of a gaint codebase like linux is still only a few hundred megabytes (uncompressed). That's more than enough data to be doing interesting work. Or look at the size of a game like quake3.