There are a couple of approaches you could take. One is silence. Rememer you’re not obliged to tell everyone everything, only those who have some kind of right to know.
For example, if you don’t want to go out it’s fine to just say ‘sorry, I REALLY don’t feel like going out that night’ without having to give a specific reason. Or you can say something that’s true but not the whole truth - ‘I have other plans’ (they may involve just you, your couch and a pizza, but hey, that’s still plans!)
In other situations you can use lighthearted ways to indicate to someone that you may have been embellishing a story a bit - you can always end with ‘… at least that’s my story, and I’m sticking with it!’ or ‘… and right after that the pigs grew wings and started flying!’
Or you might just correct yourself casually as you’re speaking … ‘it was 15 minutes later … no, I tell a lie, more like half an hour’ or something of that nature. I don’t think it’d cause much of a ruckus if you put it that way.
Of course, if it’s something that you just want to curl up and scream rather than admit to in any way shape or form, honesty is best. Start with something like ‘this is really hard for me to admit, but …’ - usually gets the other person on side, since we’ve all been there!