SSH Key Authentication on Synology NAS

Although none of the other answers directly fixed it, I took some of their methods and combined them.

This is what worked for me on my Synology DS414slim, running DSM 6.1.

I logged in as admin and executed the following command (no need to fill in your username, it will work as-is):

This will set your home directory to 700 permission, owned by [you]:users.

homedir=$(eval echo ~$USER); 
sudo chown $USER:users $homedir -R;
chmod 700 $homedir; 
chmod 700 $homedir/.ssh/;
chmod 600 $homedir/.ssh/authorized_keys;

I got it working by following tips in the comments here: https://forum.synology.com/enu/viewtopic.php?t=90208

Short version, run chmod -R 755 ~. Dir perms were too permissive (?!) for ssh to use public key authentication. No idea if this causes other conflicts, but it works.