Individual objects are split into multiple blocks, each of which can be stored independently on different underlying servers. Each can see its own block, but not any other block.
Calculating a hash like SHA256 would require a sequential scan through all blocks. This could be done with a minimum of network traffic if instead of streaming the bytes to a central server to hash, the hash state is forwarded from block server to block server in sequence. Still though, it would be a very slow serial operation that could be fairly chatty too if there are many tiny blocks.
What could work would be to use a Merkle tree hash construction where some of subdivision boundaries match the block sizes.