Has the bitcoin network ever been "down"? I'd be interested to know if the bitcoin network has ever been down. By "down" I mean that a client would not be able to synchronize with the network to obtain data like the txs in the mempool or recently mined blocks. 

My guess would be that this would be extremely rare due to the decentralized architecture, but it'd be great to know for sure. If there have been downtimes, I'd be interested to know how frequent these have been and how long they have persisted.
A bitcoin node could be down or (extremely rarely as you say) all its peers could be down... but the bitcoin network could not be considered down; you could still synchronize by connecting to different peers manually using fallback nodes or even have your own list of nodes.

The only way for the bitcoin network to be down is if all the nodes+miners are down which would be nearly impossible even with a few dozens of nodes (current estimate is 6000+ nodes). However, in the very beginning, when there where 1 or 2-3 machines it was possible (see example in @Murch's answer, where that early on, there was only one node on the network: Satoshi's).