I think it depends on what you call the reactor. I think of a reactor as the vessel that holds the fuel, moderator, controls, etc. From the NRX diagram, the reactor includes everything within the concrete "biological shield". Here's a link to the NRX reactor description:

https://inis.iaea.org/collection/NCLCollectionStore/_Public/...

The NRX was disassembled and rebuilt within a year. My guess is that Carter and others were lowered from the "revolving door" above the reactor in order to remove bolts as part of the disassembly process. The clothing worn was needed to protect the personnel from being contaminated.