I'm doing GUI programming for many years (from the DOS days) and while closures can be neat for small stuff (like updating an object field and calling some sort of refresh function for a visualization of the object when a GUI widget that represents the field changes), they are certainly not invaluable and you can do GUI programming perfectly fine without them - just pass a method pointer (obj+method) for C++ and similar languages (e.g. "procedure of object" or "function of object" in Object Pascal) or just a function address that takes an extra pointer (or whatever) argument that is given during handler registration (e.g. in a C toolkit of mine you register stuff with "register(widget, event, handler, param)" and handler is defined like "handler(widget, eventinfo, param)").
Actually, IMO if you are using closures for events that are more than 5 or so statements (i'd say lines but people love to pack their lines with multiple statements), you are doing a disservice for every poor soul who will have to understand your code later.
With enough coding, you can "do anything" in any Turing complete language. This is meaningless. It doesn't mean it is advisable (simple, maintainable) to do so. C does not support a syntax for closures, so you'll really just be calling functions with an 'environment' arg or something similar. What's the point?