the no-space-on-disc happens, if the ftp-upload is used more, than there is space on the system
FTP upload works as follows:
you connect to a server, specialised in ftp-uploading.
when you have successfully transmitted your files to the ftp-server, it will then be moved to the real file-servers, where it get's its id etc. pp
This of course will take some time.
That's why you can upload again after a few minutes (when some files were transfered to the fileservers)
as for the DDOS ... at least their main-server is hosted by a company that does have ddos-protection (though i'm not sure, if they are using big enough servers, so that's included, or if they have to pay extra)
So, yes. Nearly all file-sharing sites, have to deal with DDOS-attacks. Even uploaded was forced to buy an external service, to get their site back up - after the DDOS was done, and some time passed, they are now hosting without the service (as such services are really expensive, since they do the real deal - and not the pseudo ddos like e.g. OVH does)