It is grammatical, but the meaning is a bit obscure. The issue that seems to arise is this: somebody is (as Merriam-Webster defines it) "one or some person of unspecified identity" and anybody is "any person at all." So there's an assumption that the somebody could be a specific (possibly important) person.
"I go on longs walks each Thursday night."
"Do you meet somebody?"
"Yes, I meet Craig; we're dating."
"No, nobody's there for me." (An expectation has been denied and, to be honest, the encounter ends up being rather negative.)
"I go on long walks each Thursday night."
"Do you meet anybody?"
"No, it's quite desolate in that part of the world."
"Yes, but I don't really speak as we pass." (meet here meaning to merely pass by, as meeting a car in oncoming traffic.)
I hope that I was helpful. A lot of this is contextual, so what I've pointed out may not hold true for every sentence.
In re-reading your original post, it occurred to me that your title line is, in fact, "Did you meet somebody/anybody." I feel that the points I made still hold true of the two words, regardless of the particular tense of the verb.