Does a Chevy 350 big block exist?

No, there is no Chevy 350 cubic inch big-block engine. The Chevrolet 350 (small-block) is a very common engine, but the "big-block" Chevrolets start at 396 cubic inches (and went up to 454). There's a significant size difference between the small-block and big-block families of Chevy V8s.