What I (and I assume most people) do is to rasterize the font in an atlas at startup at different height because most of the time you know that you will only have a few sizes of text and the glyphs you need. You can change the size depending on the screen resolution or DPI scaling parameters. E.g.: at 1080p you may want text at 20, 24 and 30 pixel height, but if the user uses a 4k monitor, you multiply those numbers by 2 (or the DPI scaling factor), so you render at 40, 48 and 60 pixel height.
Note that you could rasterize each character separately (no atlas) using the actual position of the character on the screen (with subpixel precision) to get the best looking text, but if you have a lot of text that would be (I suspect) expensive. And if you use a pixel art or monospaced font, it's unnecessary.
You can also look at cleartype to have "3 time the horizontal resolution". I never used it but I think there are examples in these forums.
For storing, I just keep the font file (.ttf) in memory so I can recreate the atlas with new glyphs (mostly in application that needs to handle text input or external text source).
The part about the post processing part is just that independently of the render size of the "world" (sometimes you render at a smaller resolution and then upscale to the screen resolution for performance or artistic reasons) the UI is often rendered in a separate pass using the screen resolution so that it always looks sharp. It could also be that they actually suggest to render the font in a pixel shader but I don't know much about that (except that it's possible).