I am in a conservation area. Does that rule a garden room out?

No, but it changes the process and sometimes the design. Permitted development rights can be reduced, so an application may be needed where a house elsewhere would not require one. Materials, roof profile and how visible the building is from the street all carry more weight, which is where a fully bespoke approach helps. Timber cladding and a lower, quieter profile will usually sit more comfortably in a sensitive setting than something aggressively modern.