Not personal, per se, but the two most objectively stupid questions I've been asked where:
"What time is it?" by a guy with a fully functional watch on his wrist; and
"Do these stairs go up or down?" regarding a set of normal stairs (not escalators).
As for personal, mostly having to do with religion. I'm Catholic, and a believer, but I'm studying to be a biologist. A lot of people then go and ask "But, you're going to be a scientist, how can you believe in GAWD?". Easy, by believing. 'Course, they get angry when I answer like that >_>