www-data, the group/user Apache runs is not allowed to read in your home directory.
You can use regular permissions to change that as gertvdijk explained in its answer.
I would consider using acl instead of regular permissions, allowing to add permission to apache instead of replacing the group in home directory or making the directory world readable.
For that, you need to install acl:
sudo apt-get install acl
You can use man setfacl to have more info.
To add permissions to apache:
sudo setfacl -m d:g:www-data:X,g:www-data:X /home/me
sudo setfacl -m d:g:www-data:X,g:www-data:X /home/me/Dropbox
sudo setfacl -Rm d:g:www-data:rX,g:www-data:rX /home/me/Dropbox/Web
1st and 2nd commands will allow Apache (www-data) to change directory only through the path (without allowing other subdirectories) in your home. They are probably not needed if you are using default config but if you already changed permissions (or will change in the future) to disalow other users to read in your home, they are needed.
The 3rd one is the command that will allow Apache to read and change directories in Dropbox/web and sub-directories).
Obs: uppercase X will act only on directories instead of lowercase x that would act on both files and directories (this way apache user is only able to change directories, not executing files).
