You probably don't need the whole var folder. Try backing up just the var/[yoursiteaccess]/storage directory. The other files (caches, logs, etc) are typically not backed up.
Yeah, I know that I only need the "/var/[siteaccess]/storage", but still have the same prolem, in a complex site, the nested folders can be really loooooooooong, and cannot download them via FTP, some of them always fail...
Can you feel it Mr. Anderson? Closing in on you? Oh I can, I really should thank you after all. It was, after all, it was your life that taught me the purpose of all life. Purpose of life is to end.
The descriptions you give seem to point problems with your own ftp client application and or operating system.
I'd imagine you could use a different ftp client like say, ncftp and a unix-like operating system to retrieve the var directory recursively successfully.