The only reason that it's not feasible is because people keep saying it's not feasible.
1) You announce well ahead of time that you're not going to continue to support IE6 and IE7 (like Google does), and then when the time comes to drop support, really drop it. Don't be nice and fix "just this one issue, I promise". Legacy browsers need to die.
2) Yes. If it's for legitimate work purposes, then they can either ask for another browser to be installed, or ask for their current browser to be upgraded. I personally was responsible for the upgrade of thousands of computers from IE7 to IE8, simply because a web app that my group needed access to didn't support IE7.
And FWIW, I'm not saying "don't browser test". I'm saying, maintain a very strict set of browsers that you're willing to test on, and don't include legacy browsers. A very good approach is the most recent two or three versions of IE, Firefox, Chrome, Safari, and the stock Android browser.
