all eMMC /dev nodes seem to be there. which is good.
/dev/mmcblk0p1 is your MyDocs, and /dev/mmcblk0p3 is swap

My advice would be to:
1) install BackupMenu from the repositories.
2) use it to backup rootfs and optfs to emmc or uSD (if it will succeed mounting them) OR
-- try to use Raw USB feature of BackupMenu, which will mount all the partitions onto your pc
3) After you've backed up your stuff (maybe more than once to different places), use BackupMenu (or PC) to run 'fsck' on the partitions.

If you're lucky, errors will be fixed and they should mount.

Obviously, the fastest way (but least desirable) is to reflash your device.