Qwerty, you're not a stupid person, don't make a stupid person mistake.
1.) I dated a co-worker who was roughly the same level as I was, maybe slightly lower than me, but that's not relevant. When things fell apart, it was a mess in every sense of the word. You want to know what a rumor mill is first hand, date your boss. Your co-workers will slowly resent and push you away, you have to hide your relationship for conflict of interest reasons, and when it collapses the fallout invades every corner of your life.
2.) You think that this can't potentially harm you career, but no one plays by the rules. For instance, a company can't comment on your performance while you worked for them, legally, but they can say "I would not hire xxxxxxx again." Labor laws are littered with loopholes.
I was fired from my previous job and was given no concrete reason and still to this day am not clear why it happened. The girl I dated, whom I wouldn't even acknowledge, my entire department and sister departments, two assistant store managers and my direct supervisor sided with me. My store manager didn't think I would peruse anything, but I went after them and won compensation and the legal half of it. Businesses will bend you over if you pose any problem for the company. They have two goals, to survive, and to be profitable/for shareholders. Nothing will stand in the way, it doesn't matter how much so and so likes you, or how high up your friends are.
Not only do I think this is a horrible idea from your end, its very irresponsible of your boss to even more in this direction. That's kind of a red flag on its own in my eyes. You're a cute......and....wait for it.......intelligent girl, you can do MUCH better.
Does your boss look like me?