Since conventional computers use bits, which can either be 1 or 0, can we simulate a 64-bit operating system on a quantum computer by restricting the values of the qubits to either 0 or 1, instead of 0, 1, and everything in between?
Yes, in principle it's possible to simulate any classical computation on a sufficiently large quantum computer.
Any deterministic classical circuit can be implemented as an equivalent quantum circuit using Toffoli (CCNOT) gate - it can simulate classical gates NAND and FANOUT, which are universal for classical circuits.
Quantum computer can also simulate non-deterministic classical circuits by generating fair coin tosses using Hadamard gates followed by measurements (which gives 0 or 1 result with 50/50 probability).
So with the proper software you can simulate any classical computation. However, running a classical OS on a quantum computer would be rather pointless, since classical computers do this quite well already.