It probably wouldn't be usable for a general-purpose programming language, but for a special-purpose scripting language I could see it making the language implementation easier.
In general, I think that cannot be done, but if one restricts what programs can do, solutions exist.
A simple way to do it is by requiring all references “pointing out of” an object to be set the moment the object is created, and be immutable afterwards (that’s what Lisp cons (https://en.wikipedia.org/wiki/Cons) does. Without setf or similar, lisp code cannot create cycles)
That disallows quite a ome code that modifies structures without introducing cycles, but still allows for quite some code to work.
One could also store an ‘age’ field with each object and check, when a reference is updated in an object, that it points to an object that is older than the one being modified. That gives some more leeway, at the price of using more (a lot more, in code using small objects) memory.
Another idea is to add a bit to each object “there are no cycles containing this object”, and have the runtime clear that when it no longer can guarantee that (edit: unfortunately, maintaining that invariant can be very costly. Whenever code does foo.field = bar, with both foo and bar known to be not part of a cycle, you still have to do a search through all objects reachable from bar to check whether a cycle was created and, if so, clear that bit in all objects in the cycle(s). That makes this idea impractical)
If, as I suspect happens in programming languages which are “mostly immutable”, there are many objects for which that flag stays set, that can significantly speed up checking for the creation of cycles.