In the case of Stamen, the process consists in installing Tilemill (the project is originally from Mapbox but it's been forked because Mapbox has changed its rendering/cartography technology since then: https://github.com/tilemill-project/tilemill) and then cloning the Stamen project (https://github.com/stamen/terrain-classic). From there, you can follow instructions, tweak the Makefile, etc. The source of the data is OpenStreetMap and NaturalEarth so even if you don't want to change the theme you may at least update the data sources.
The process is quite complex, you'll also need a PostreSQL database with the GIS extension installed to ingest data from OpenStreetMap. A quite powerful server and a lot of RAM are required to be able to do the rendering efficiently.
Unless you need to do that, I wouldn't suggest trying. It's a nightmare with several sets of contradictory, overlapping out of date docs needed to do the whole thing properly. Then of course it breaks and you don't know why. It used to be that tilemill kinda nailed it but that just went away.
