The space should be 1GB, the installation with all uploads has 75MB. File upload via FTP does work.
Edit: I contacted someone at the webhoster. He said that the working FTP transfer is an indicator that this is probably a CMS related problem, not a server problem. Any ideas?
Edit 2: I just noticed that this happens only with the fields in one specific template.
Edit 3: The problem were special characters in file names.