How do you lower yourself down into a reactor that has melted down at thousands of degrees and come out alive or not suffer radiation poisoning? The story seems embellished.
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.