So what exactly does one do when the entire team is feeding to "carry" them?

First off, some games are unwinnable. Not anywhere near all the games you (or I) lose are, but some are, so don't beat yourself up for losing a game that was a lost cause. That said, I've won many games with a 2/13 bot lane or the equivalent, and I guess there are a couple things you should be aware of? First, it depends a lot on what champion you're playing. I prefer playing mid lane burst mages because of their carrying potential; top lane bruisers or champions named Vladimir can do the same thing. Other people may have tips of their own, but as far as I'm aware, if you're support/immobile adc/pure tank top or jungle, you are pretty much unable to solo carry. You should always try and focus the most fed damage-dealer on the enemy team, and during laning phase you should try and have as much of a global impact as you can. Roam or use Teleport aggressively, because even if you snowball your own lane, it's equally important to try and slow down the lanes that are feeding so that the enemy team doesn't snowball as well. Look for picks and capitalize on mistakes, because lots of people are really really bad at this game, including me. That said, do not be stupid. Don't engage on the guy you see walking through their jungle without vision to know where their team is because you want a pick or dumb crap like that; that just puts you at risk of falling further behind. Honestly, the most important thing is to stay focused, don't tilt, and chat often! Call out your team for minor good plays or picks that they make, and don't flame them for mistakes. It sounds dumb, but I swear it's been a huge part of my ongoing climb to Plat; you need to keep you and your team off tilt. Even saying dumb stuff like "we can still win this" after a decent trade is helpful. And, like I said, you're still gonna lose a lot of games, because we all suck at League of Legends. But hopefully you can lose a few fewer :P