There are two ways to store map tile data like OSM. One is as pre-rendered raster tiles (256x256 png images) and the other is as tiles of vector data that get rendered either on the fly by dedicate rendering middleware or rendered client side by the browser. Vector tiles are anywhere between 20-50% smaller that corresponding raster tiles. Raster tiles was the standard way of doing web maps up until a few years ago, when client side rendering in the browser became feasible and now most people use vector tiles.
Yeah going to full 20 requires about 2TB iirc, OSM doesn't host beyond level 19 anymore either to reduce traffic. 14 is around 10m per pixel, best you'll be able to make out on that is like a street. Not useful enough to bother with imho, even 19 leaves a lot to be desired up close.
Worth noting that these are vector tiles and not raster tiles, so zoom levels work slightly differently. Generally you don't need as high zoom level with vector tiles since it is much easier to 'zoom in' on vector data than on raster data. A vector zoom level of 14 stores the data with roughly 50cm precision which is good enough for most casual mapping uses.
