Facebook Questions has relaunched as a limited public beta. The limited means it is meant to spread through your personal network: when your Facebook friend asks a question, you can start using the service. The "public" part means that you can bypass that by simply clicking on the green Go to Questions button.
See Emil Protalinski's report, Facebook Questions relaunched as limited public beta.
You're in! After you click on the green button, the text under it will change from "We'll be rolling out Questions to everyone soon, but you can upgrade your account now." to "Your account is already enabled for Questions."
When you click Okay, you will be redirected to the main Questions page. Here's what it looks like if you have no friends.
Here's what it looks like when you do have friends: the page will be full of questions answered by users on Facebook, sorted chronologically. The questions that your friends posed and/or answered are given a higher priority.
If you highlight one of the answers, you will be told the exact number of votes it has received. More importantly, you can highlight the ... part on the right to see which of your friends voted for what answer.
At the top of the page, click on the Question button to pose your own question.
Write your own question and your own poll options. The "Allow anyone to add options" checkbox is checked by default but you can uncheck it if you want users to be limited to your options.
Notice that possible poll options can be Facebook content. When you start typing, Facebook gives you a list of possible pages you may want to link to.
When you hit the blue Ask Question button, you will be prompted with this warning.
Here's the final result. It appears on your friend's News Feed instantly.