Does a storage room in a Chinese take away require a fire door when the room is set back from the kitchen and is not within the means of escape 

Carry out a suitable and sufficient fire risk assessment to establish where fire doors (if any) are required.

Guidance at: https://www.london-fire.gov.uk/safety/the-workplace/takeaways-cafes-and-restaurants/   and    https://www.gov.uk/government/publications/fire-safety-risk-assessment-offices-and-shops  for example.

Depends on the risk of fire at the location, the assessment could include (but the list is not exhaustive):   Is there possible risk at the storage room? Are flammables stored there / is there risk of ignition?   What electrical equipment or installation is there at the location?  If there was a fire at the location, how would people at the building be affected?  Is there any sleeping accommodation?  Are there detectors and/or other fire safety devices?  Would a fire at that location affect ease of escape?