Not only did some 1960s engineers at General Electric think that this might work, but they did actual tests involving actual hardware. NASA and the USAF declined to pursue the project, for fairly obvious reasons.
Sputnik really freaked out Americans who believed in American exceptionalism. I wonder what other harebrained space stuff that's unknown to the public due to "national security" the US was involved in the early 1960s. "National security" actually meaning "national pride" because even 60 years later - it would be embarrassing for the public to know about it.
I love how Wikipedia paraphrases the concept - "This plan was batshit."
The MOOSE system was nonetheless always intended as an extreme emergency measure when no other option for returning an astronaut to Earth existed; falling from orbit protected by nothing more than a spacesuit and a bag of foam was unlikely to ever become a particularly safe—or enticing—maneuver.
What did MOOSE stand for?
MOOSE, originally an acronym for Man Out Of Space Easiest but later changed to the more professional-sounding Manned Orbital Operations Safety Equipment
Given how so many other truly batshit-crazy projects and proposals from that era are public knowledge, it would have to be something really embarassing.
Another classic that also went through preliminary hardware testing was Project Orion. It has nothing to do with the current Orion capsule being developed for launch on the SLS. It was a proposal to propel spacecraft using the shock waves from a series of nuclear bomb explosions. And yes, there's no air in space, the shock waves would be made up of the material of the bomb casing itself. Engineers actually built small-scale mockups using conventional explosives and put them on test stands.
Honestly this doesn't sound that terrible as the equivalent of an emergency ejection seat.
Developed to the point of reliability it probably would be a nice option for a few things. Like space stations don't have room for much, but this is small and low enough mass to boost up a few and have stowed.
Imagine a medical emergency, no recovery rocket available or no time. Less an issue nowdays rockets are getting more numerous and cheaper but that wasn't a given. Your choices are die in space or probably live. The foam might even help stabilise any splinted limbs or whatever.