you could crack a 768-bit RSA in... roughly guessed... ...a third of a day.
Sorry, no. That doesn't take into account the fact that some parts can't be run in parallel on many home computers. Not to mention that the longest part, sieving, for a number this size, needs about 1 GB of RAM free, which I'd think people would be likely to notice and shut down pretty quickly...
Sieving is the step that takes the most time, in this case 1500 CPU years ("On a single core 2.2 GHz AMD Opteron processor with 2 GB RAM per core, sieving would have taken about fifteen hundred years."), but can easily be run in parallel. Let's say you have access to 100,000 cores, each with at least 1 GB of RAM that you can use (read the PDF...). It will now take you 5.475 days to do the sieving.
Polynomial selection can, like sieving, be easily distributed, and is a relatively trivial task with 100,000 cores available. (roughly 20 CPU years, or under 2 botnet-hours, and a non-enormous amount of RAM)
The hard parts are the final steps: filtering, building a matrix, solving it, and finding the factors. You basically need one or more supercomputers to do it, with at least one of them having 1 TB of RAM and fast access to 5 TB of data. To do it like they did, you'd also need to write your own block Wiedmann implementation. If not, you'd have to use the block Lanczos, which can only be run on a single computer/supercomputer/cluster.
Doubtless, someone could botnet enough computing power to sieve for an RSA-768 key in a matter of weeks, but to actually finish it and get the factors would require an expensive supercomputer, be it purchased, (better hope whatever's behind that key is valuable...and thank goodness that they were stupid enough to use just a 768-bit key on it) botnetted, (good luck to get one and not have anyone notice!) or otherwise acquired.