He never carries all six at once, except when they're combined into one. The weight would be ridiculous, of course, assuming the swords are steel - which they quite possibly aren't. There's plenty of fictional material in fantasy worlds. After all, FFVII itself included Mythril and Materia.

The complete six-part sword isn't that much bigger than the original Buster Sword. Even before he got his Jenova enhancements, Cloud was able to wield that one - albeit clumsily - and wound Sephiroth with it, at the Nibel Reactor. It seems fairly clear that those utterly colossal swords are only meant to be used by warriors with truly superhuman strength.