Should I say "data sets of race times of various cars" or something better? I am concerned that the "of" don't sound quite right.
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In many cases you could just say "sets of car race times".
In the original phrase, the word "data" is redundant, since care race times are obviously data. Substituting "car" for "cars" lets you eliminate the "various".
If you want to keep the original structure, you could say "sets of race times for various cars".
It's an unwieldy, but grammatically correct, way to put it. "[D]ata sets of cars' race times" could also work. The problem isn't your phrasing, but that dealing with multiple sets of data about multiple cars with (at least one) race time per car is itself a little unwieldy.
If you have to talk about it a lot, give it a memorable nickname or acronym.