In the original Star Trek TV show (TOS), Scotty and his engineers faced a galaxy-load of hazards from a warp-core melt-down to a phaser overload. Radiaition was constantly mentioned as an ever-present threat as matter and anti-matter annihilated each other inside the ship's massive power core. And yet, despite this much-ballyhooed danger, Scotty and the boys did very little about it. Of course, it's no secret why Engineers wore no protective clothing – it was all about expense. In the real world, those that worked around radiation needed to wear some type of protective suit. But Star Trek couldn't afford that type of realism, so its workers wore red uniform shirts and apparently hoped for the best.
But then came Star Trek: The Motion Picture, and suddenly the budget didn't matter. Now there were special uniforms for every possible need, including cool new Protective Radiation Suits or Rad Suits. These would be seen in various Engineering scenes as Scotty's crew worked on keeping the Warp Core in one piece.But they were strictly background pieces, meant to flesh out the overall sophisticated look of the film.
Then Wrath of Khan came along and suddenly these Rad Suits were not only heaviliy used but they were shown as being literal life-savers as the Enterprise fought the Reliant. Suddenly, Engineering was brimming with explosions and unchecked radiation. The Rad Suits became part of the story-telling, showing us that even with such protection, death was ever-present in the unknown reaches of space.
The Rad Suits featured heavy boots modified from an Italian-made boot, along with heavy gloves and various padded details meant to represent protective materials and surfaces. They had various clips and loops for tools as well as details that might represent some type of radiation detectors or exposure dosimeters. A Starfleet patch was placed on the shoulder and a rank strap was attached at the wrist in the style of all Wrath of Khan uniforms.
In Star Trek VI, the Rad Suits would play a crucial role. They would be worn by two Starfleet assassins who beam over to the Klingon ship and murder Chancellor Gorkon and several of his officers. They would be modified with the addition of a helmet (to hide the assassin's identity) and a pair of Gravity Boots to help them navigate the zero-G environment of the powerless Klingon ship.
It's interesting to note that in later Star Trek incarnations like The Next Generation or Deep Space Nine, no Rad Suits are shown. Again, the budgets of TV story-telling required things to be simplified. The Rad Suits went away, never to be seen again.