Hi,
Why using RAID / why choosing RAID6?
so I suspect it is recoverable
Nope, R6 with three missing disks is not recoverable ... but may be you'll get one more member running.
but have no idea where to begin.
List your hardware (some details about the mainboard and/or the sata-controller, and of course the HDDs used) ... then look into the log messages of your system, eventually you reveal some hints about the errors ...
The only chance you have (before backup :P), is to recover on or more superblocks from the RAID - it is a common bug, that this superblocks are not written "physically" to the media - and getting lost during reboot (or powerloss), but this will be very difficult ...
Did you read https://raid.wiki.kernel.org/index.php/RAID_Recovery ? Or any other RAID-recovery-related article, which handles with more lost drives than redundancy has? (which ones?)
Start with playing around with:
mdadm --examine /dev/sd[...] may be you'll find some backup-superblocks ... otherwise your data is gone
Sc0rp