The largest airplane possible is a function of time. At any point in time, the then current technology will have a practical maximum size due to square-cube effects mentioned in this answer:
However, technology advances. The C-5 was developed in the 1960s, with a MTOW of 417 ton. Compared to the 142 ton of the then current 707-320, this was a groundbreaking engineering effort at the time. And no, mounting more engines also runs into problems at the then current technology, like the Spruce Goose experienced, which did not fly out of ground effect.
Did they mean the c 5 was actually the largest physically possible aircraft or just that anything larger would have weakening service life?
Lifespan is indeed one of the design parameters for size, however the C-5M is a C-5 with upgraded engines and avionics according tho the wiki article, so some of the lifespan of the original planes was already consumed. The An-225 with its MTOW of 640 tons was developed in the 1980s.