You do not define the meaning of the word “safe”. That question is, therefore, not answerable. (safe can mean safe from deletion, safe from corruption, safe from unauthorized access, safe from law enforcement, safe from in-transit capture, safe from prying eyes, safe from system administrators, etc. – and varying levels thereof, i.e. “safe from unsophisticated webbrowser” over “safe from password-guessers” to “compartmentalized top-secret kind of safe”.
It’s feasible. Don’t use it as the ONLY storage for those files. You will regret it some day.
Uploading 20gb over DSL can take a while, depending on how fast your DSL line is. “DSL” can mean many things. ADSL, SDSL, VDSL, ADSL2+, etc., in varying speeds. With 1mbit/s upstream, for instance, it’ll take about two days to transfer the files there. With 512kbit/s upstream 4 days, with 4mbit/s upstream half a day.
You may want to consider using SFTP, ssh+rsync, or scp to do the transfers to ensure the security of the connection from you to Dreamhost. (WinSCP is a nice tool for that under Windows, for instance).
I have about ~300gb files on my dreamhost account, most of which being backups from elsewhere. If you require confidentiality, have a look at duplicity; It combines rsync with strong encryption via GPG, so unless you have the key, the files are worthless. (And no, this isn’t “just” useful for nefarious purposes; there is plenty of legislation requiring protection of privacy and data in various jurisdictions; not using encryption for off-site backups under the control of another party (DH in this case) just may be criminal negligence; even if that does not apply to you and you trust Dreamhost staff completely, security breaches happen everywhere. Even at DH ;).
If confidentiality is not that big a problem, you could just use regular FTP or WebDAV folders to do this; some imaging software allows you to use WebDAV as a storage medium, as well; and some operating systems allow you to mount WebDAV sources as drives (MacOS makes this easy; Windows has some broken support for it, though there are decent third-party clients from Novell, for instance).