Would it be possible for me to have a wedding at Saint Peter’s Basilica? Would it be possible for the Pope to officiate? How much of a problem is it if I am not Catholic? Would I have to convert? What if I wanted to marry a woman who is divorced (another problem?)?
Actually, yes, it is possible I believe to marry at the Basilica-and surprisingly inexpensive. I read about this years ago, I wish I could remember where!
However, there are a whole ton of rules, and if you’re not Catholic, and your bride to be was divorced-then no, it wouldn’t.
And no, the Pope wouldn’t be doing it. Especially now.
Moderator’s Note: Even though the OP mentions the Roman Catholic Church, the Pope, and divorce, I don’t see why the questions in this thread can’t be given factual answers, so I’ll move this over to GQ in case there are any factual additions or emendations anyone wants to make to Guinastasia’s post.
Here’s a tour operator who does this: http://www.weddingatstpeters.com/
They have a whole page on eligibility requirements. Only one Catholic = ok, divorced = not ok.
Actually, it says if you’re divorced, you can get married at Santa Susanna Church, in Rome. That’s odd.
Note that you’re not getting married in the Basilica. You’re getting married in a nearby chapel. I looked for a picture but found nothing.
Maybe the “Santa Susanna Church” wedding is just a civil affair carried out in a nearby courthouse? 
One of the requirements for being a Basilica is that the church must contain a certain amount of chapels within. So you’ll be inside the Basilica, specifically in one of the chapels.
Yeah, I realised my post wasn’t so clear. What I had meant to say was that you shouldn’t expect to get married under Bernini’s Baldacchino in the main nave.