ANy distro with Opera, you can then use the built in Opera bittorrent client.
MeanPuppy Linux has opera as Standard.
This is also cool. (just needs Ktorrent or Opera)
http://damnsmalllinux.org/store/Mini_ITX_Systems/Damn_Small_MachineYou can run the entire thing from a USB key as your harddrive.
I have seen it somewhere done on a cheaper machine though.
Motherboard, processor and USB key.
hard drive not required.
(saw one for $57 US)
Saying that, when I was in New Zealand I just used to remote into the Local net cafe, que up my downloads and pop in to pick them up every few days.
$1 NZ Dollar an hour with about 3-4 days downloading between visits
