MySQL server has gone away

You will need to check with your host see if you Mysql server problems. In phpmydmin see if you still have your database on server.

Switch to default theme and see if that is working

You mean you can not log in wordpress admin if so using ftp log in and on right box browse to wp-content/themes folder in themes folder you will have your theme rename that theme like add -hold after the name.

looks like you have permission problem where do have your wordpress files and folders are they in wp or wp-oud folder

may be you have the urls wrong in Settings>>>>General as they should be http://nautilus-amsterdam.com/wp
where did you get
http://nautilus-amsterdam.com/wp-oud/ from.

I get 500 error when I go on http://nautilus-amsterdam.com/wp and all your links on http://nautilus-amsterdam.com/wp-oud/ are pointing to wp so there is problem your setting up.