The faulty partition is /dev/sdb3.
from /proc/mdstat:
md2 : active raid1 sda3[2]
2896184639 blocks super 1.2 [2/1] [U_]
says /etc/fstab:
# /dev/md/2 belongs to LVM volume group 'vg0'
/dev/vg0/backups /backups ext4 defaults 0 0
/dev/vg0/root / ext4 defaults 0 0
says pvs:
PV VG Fmt Attr PSize PFree
/dev/md2 vg0 lvm2 a-- 2.70t 50.00g
says lvs:
LV VG Attr LSize Pool Origin Data% Move Log Copy% Convert
backups vg0 -wi-ao-- 974.00g
root vg0 -wi-ao-- 1.70t
Is it possible to fsck.ext4 (or badblock) the /dev/sdb3, as long as the machine is running? (W/out getting a "Device or resource busy" due to the LVM).
You shouldn't even try to do fsck on a specific partition, because you're using software RAID and LV on top of it. So the filesystem is on other/"higher" level.
But you could use badblocks on a specific partition if it's not used by OS (/dev/sdb3 in this case). Then you could (and you should) re-add your partition to the RAID, if the device state is OK, by using option "--re-add".