After several successful years, this group disappeared from their skyscraper headquarters and reappeared in Latveria as bodyguards to Dr. Victor Von Doom, the monarch of that little country. They were called the Doom Squad and seemingly never aged (nor is there any evidence their leader had, either). The Baxter Building penthouse complex became the Latverian consulate in New York.
In 1996, on the trail of Annihilus's cosmic rod which had been stolen from them by the Doom Squad, ISIS invaded Latveria and discovered that the F4 were in fact being held in a stasis while their life forces motivated robots of themselves under Dr. Doom's control. Once freed they initiated procedings to evict the Latverian embassy and took back their HQ.
Apparently they were aware of what was going on but had no control over their actions as far as obeying Dr. Doom (who apparently was also an impostor). The stasis tubes they were held in apparently retarded their physical aging, they had not aged appreciably since they disappeared 26 years ago. Reed Richards spent two years speed-reading his way through 26 years of scientific journals. For awhile, rumor had it that they might not take up the name Fantastic Four again, but they eventually decided to retain it.