Downloading a file from the Internet in Standby Mode?

"No, you can't. When the computer is in standby mode it is off, except for the memory being constantly refreshed (if you don't refresh the memory it clears).

If you want to perform network activity the computer must be on so that the CPU can execute application instructions. If you want to save power I suggest turning your monitor off or setting power-saving settings in your operating system (such as auto-monitor standby after a period of time).

Note that the application you suggested will probably also want to access a hard drive or other storage mechanism."

"A solution would be to buy a NAS (Network Attached Storage) that supports BitTorrent.

For instance, QNAP makes such NASes. Apart from providing disk space to computers attached to the LAN, the QNAP NAS boxes have a lot of other features, including:

Download via HTTP using the built-in Download Manager
Download via Bittorrent using the built-in Download Manager, or the add-on rTorrent+
Download on Usenet (newsgroups) using the add-on SABnzbd+
Download on P2P networks using addons like MLDonkey
...

They use Linux and can also run PHP, MySQL, websites, ...

I bought the Qnap TS-210 and I'm very satisfied with it.
Electricity consumption is around 14W with 2 disks in operation."