Yes, it is possible to emulate a quantum computer, and this is done quite often for research and to try out quantum programs.
The problem is it takes O(2^n) classical computer resources to emulate a general purpose n-qubit quantum computer. In other words, exponential time or size.
(We can simulate some larger quantum chemistry systems on a classical computer, but those aren't general purpose. The simulations are quite restricted in what they can measure, and there's still a significant practical size limit.)
So we can only emulate very small general purpose quantum computers or other quantum systems. For larger quantum computers, in principle those can be emulated too, except you would need an impossibly fast and large classical computer to do it. So we can't do so in practice.
This is actually the motivation for building real quantum computers of significant capacity.
If a real quantum computer can be built with a large number of high quality, fully coherent qubits, it will be able to do calculations that can't be emulated on any classical computer we can actually build and run, just because of the O(2^n) practical limit.
Right now, there are no quantum computers like that. There are some dubious marketing claims around, and there are also some genuine, but smaller, devices.
Because we can't even simulate a large quantum computer, we don't know for certain whether such a device can even be built in the physical world. The abstracted maths of quantum mechanics, which has proven to be extremely accurate and correct for everything it's been used on, says it can (subject to practical engineering details), but the physical world may have a subtle limitation which we can't detect in smaller systems, that only happens with larger quantum computers and prevents it from being possible. The maths itself might even have a subtle reason (such as stability or entropy) why the system cannot work, but no such reason is known at the moment. We can't "run" the maths to find out its behaviour on a large system, for the same reason we can't simulate a large quantum computer without a large quantum computer in the first place. We can only reason about it in the abstract.
The problem is it takes O(2^n) classical computer resources to emulate a general purpose n-qubit quantum computer. In other words, exponential time or size.
(We can simulate some larger quantum chemistry systems on a classical computer, but those aren't general purpose. The simulations are quite restricted in what they can measure, and there's still a significant practical size limit.)
So we can only emulate very small general purpose quantum computers or other quantum systems. For larger quantum computers, in principle those can be emulated too, except you would need an impossibly fast and large classical computer to do it. So we can't do so in practice.
This is actually the motivation for building real quantum computers of significant capacity.
If a real quantum computer can be built with a large number of high quality, fully coherent qubits, it will be able to do calculations that can't be emulated on any classical computer we can actually build and run, just because of the O(2^n) practical limit.
Right now, there are no quantum computers like that. There are some dubious marketing claims around, and there are also some genuine, but smaller, devices.
Because we can't even simulate a large quantum computer, we don't know for certain whether such a device can even be built in the physical world. The abstracted maths of quantum mechanics, which has proven to be extremely accurate and correct for everything it's been used on, says it can (subject to practical engineering details), but the physical world may have a subtle limitation which we can't detect in smaller systems, that only happens with larger quantum computers and prevents it from being possible. The maths itself might even have a subtle reason (such as stability or entropy) why the system cannot work, but no such reason is known at the moment. We can't "run" the maths to find out its behaviour on a large system, for the same reason we can't simulate a large quantum computer without a large quantum computer in the first place. We can only reason about it in the abstract.