Possible alternatives are:
I haven't had a moment's rest 249 results
I haven't had a moment's peace 2,360 results
I haven't had a break 1,650 results
The first idiom is less popular but in my opinion describes more accurately the idea of working 24 hours a day.
Not to have "a moment's peace", on the other hand, is slightly ambiguous. It could also mean you were stuck in a noisy situation for hours on end. Peace in that context makes me think of: "peace and quiet".
The last idiom is again ambiguous, left alone it could mean you haven't had a holiday or a weekend off in a long time. "I haven't had a break all day" works well.
I've been on my feet (all day) 15,200 results, sounds more natural than "I've been running all day" as Tristan pointed out, it could mean a sports person training fanatically.
Nowadays, I suspect many might say: I've been busy twenty-four seven (24/7).
It's not an expression I ever heard in my childhood, and not one I personally use, but it is unambiguous.