, the M14 went into service 4 years after the Korean war ended, the relic you are thinking of must me the excellent M1 Garand.
, those bullpup conversion kits are pretty slick. There are two kits available, BullDog 7.62 and the Rogue.
This was my Rogue with an 18" barrel.
The piston driven M14/M1A is probably the all around best 7.62/3.08 battle rifle design available.