I refrained from saying why in order to focus on what I need now. However, since you ask, the eviction notice began: "Due to multiple server overloads and based on our terms of service and hosting agreement we have cancelled your hosting account with us."
My little blog only got a dozen or three visits per day, and is light on graphics, so I doubt that the traffic was more than 10 or 20 mb per day. Of course, since the account has been cancelled, I cannot see whether maybe something happened yesterday to cause a problem.
In a subsequent message, my former host said "We are sorry for the inconvenience caused but we were pressured by the other inhabitants of this server to either stop the overloading or they would cancel their contracts." Not very sorry I guess, as they did not even offer to upgrade me to a more costly option. This particular host uses an obscure measure, "percent of processor" or something like that, and provides no warning when one is approaching a limit. It's really quite strange, as their support was pretty good during the time I was with them, even when I screwed up the site all by myself they explained promptly and clearly what I needed to do to fix it.
What they sent me is a tar.gz file of 438 mb, which when decompressed is something over 500 mb, with a couple dozen directories including a mysql directory of something like 50 mb. Presumably I have the complete site, including images. I also did regular backups using WordPress tools, but these are much smaller, 1.9 mb. I don't know what's missing from them.
A "more traditional move" might be the way to go, but it's not "traditional" for me as I've never moved a WordPress blog before, so I don't know how simple or difficult it might be.