Sorry for the delay but I've been (and I am) a bit bussy
Yesterday I check your domain and I saw that it was served by a web server named "web station" or something like that and you were redirecting it to yourdomain:5050. Today it is served by nginx and it does not redirect to any other port so don't know whether this nginx server is a proxy server redirecting request to another machine or it is a web server running in your Synology NAS. Also, don't know what is the default web server used by your NAS nor if syno-letsencrypt can work with it...
Anyway, I don't use Synology so I've no idea how it works but, the first thing you should check is that the challenge needed by Let's Encrypt is reachable from internet. So, you should check what is the document root used by your domain (you can check it in your nginx conf and you should see a root directive in the nginx file, something like this:
In this case the document root is
/usr/share/nginx/html so you should create the following directories
.well-known/acme-challenge/ inside your document root.
mkdir -p /usr/share/nginx/html/.well-known/acme-challenge/
Once you have created the dirs (keep in mind that /usr/share/nginx/html/ is just an example and you should change it for the right path) you should create a test file with whatever text.
echo "This is a test" > /usr/share/nginx/html/.well-known/acme-challenge/test
Now you should check that you can get the file with your browser: http://backup.fms-data.dk/.well-known/acme-challenge/test
If you get
This is a test then this part is working fine and we should focus on how
syno-letsencrypt works but if you don't, you should review your web server conf.
@serverco requested it to see the perms of syno-letsencrypt but there was a typo in the command he wrote so you are viewing the inode number used by syno-letsencrypt instead of the perms. It should be
ls -l /usr/syno/sbin/syno-letsencrypt instead of
ls -i /usr/syno/sbin/syno-letsencrypt
Again, sorry for the delay but I don't have too much time to answer these days.