The way Factom works, they store the merkle root of the various hashes on the blockchain in an OP_RETURN output. This amounts to 80 bytes in the OP_RETURN and the actual transaction is going to be on the order of 500 bytes. The way they scale is that they store the actual merkle tree elsewhere and only put the root into the blockchain, about once per block.
Assuming that everyone uses Factom, this would mean that the blockchain would only get one Factom transaction per block, which is what it's already doing. If there were a million other services using OP_RETURN, then no, currently, the block size is limited to 1MB per block, which can only take about 2000 such transactions per block.
1It's, by definition, not an "alternative chain" (it can't do without the Bitcoin blockchain, see answer). More like a "dependant chain" or "child chain" or something like that. So it's not actually Factom that is doing the timestamping (it would need Proof of Work to do that), it uses Bitcoin to do the timestamping. It simply bundles a lot of facts into one hash that gets timestamped on the Bitcoin blockchain. – Jannes – 2015-12-09T19:31:44.467
@Jannes: took out the "alternative" part. – Tom Au – 2015-12-09T21:09:04.283