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Yes, theoretically of course it is.
Even practically it seems, they are all over the place.
They are not. Realistically this is not possible. And may never be.
Complexity is a real problem here. The more different things you want your machine to do, the more human intervention is necessary.
Let's give this idea the best chance at success and say it's raw materials and simple components are available to it. Basic fasteners, copper wire, sheet metal, etc.
This robot will undoubtedly have a motor, which would require a bunch of copper wire wound around a magnet, and assembled in a ring. This task is already done by robots. At least, the winding part is. The assembly is too, but it's a different robot. And let's say there's even a robot that can transport the wound magnets to the assembly robot. Such robots exist at least, this wouldn't be logistically efficient in this particular process but we're building a self replicating automaton here, sacrifices are necessary.
Anyway this one task requires at least 3 robots AND at present, the raw materials are being initially fed by a human. The magnets might come from a hopper but the copper wire is an issue. Let's say even THAT is surmountable.
The amount of floor space required for the process is probably something like 300 square feet. Just this process alone. Just to make a single type of electromagnet. We haven't built the housing, inserted the drive shaft, or any of the other required bits. just the electromagnet.
To make all the parts of a thing that a robot requires is going to be about the size of a factory. and that's assuming raw materials and basic bits are made somewhere else.
And even if you fully automate a factory, and it can make everything it needs to replicate itself, different machines would be needed to install them all somewhere.
You're thinking "but 3D printers!"
Stop. No. Just... No. are not that powerful. To be clear, a single metal 3D printer that's even half as capable as some people have been lead to believe they are could replace about 20 of the highest paid employees in my factory. They just don't work as well in industry as people seem to think. And that's printing in a single material. When you try to say, print a magnet with copper wire wrapped around it, you find very quickly that a: this isn't possible with the current technology, and even if it were, b, the durability just isn't there. It might be hard, it might be the right amount of brittle, but it doesn't withstand wear the way you'd expect for something with those properties.
My response was a bit tongue in cheek, I was hinting at Cells.
So theoretically it's absolutely possible, it exists already.
You could even see humanity as a (not human created but still existing) a form of self replicating machinery.
For a man-built one, you might want to check out Dr Drexlers book Engines of creation (or is it Nanomachines), where he makes the claim that it is indeed possible. He lays out the theoretical math & physics to create APM, atomically precise manufacturing, and that machinery would indeed be able to create a perfect copy of itself.
It's a chemical and engineering problem. Plants and animals do this already. They are our examples to learn from.
A synthetic plant is a machine.
The question was about theoretical possibility - practical limitations of present-day tech are irrelevant. OP also didn't put any constraints on how big, heavy or complex the robot would have to be, or if it has to be able to do anything else besides make more of itself. Even if the machinery it has to contain occupies a city block, that's okay.