It's more practical than you may think. Just needs about 40 GBs right now. I did it a couple years back in a fit of peculiar paranoia, downloaded the full hash list and checked all my KeePass-stored passwords at that time against it.
The above post https://news.ycombinator.com/item?id=45840724 links to 71.3 KiB of data; since it's a 5-nybble prefix (20 bits) we may easily estimate a size of 71.3 GiB assuming that's a representative sample. Not unfeasible nowadays, but it seems you do have to make separate requests and would presumably be rate-limited on them.
If you only download the hash pages corresponding to passwords you hold, even supposing that everything else is fully compromised, an attacker would have to reverse a couple thousand SHA-1 hashes, dodge hash collisions, and brute-force with the results (yes, yes: arson, murder and jaywalking) to pwn you.
