We can store such information in a 2-level quantum system. For qubits, we require that we can initialise a state with high fidelity, manipulate it according to some sets of unitary operations, and that we can read out the state. Many different types of quantum systems can be used as qubits:
- Photons (where polarisation spans the computational basis)
- Superconducting qubits
- Neutral atom qubits (where metastable hyperfine electron states carry the 0 and 1 information)
and many more! The information is then stored in the system itself, which will exist with good fidelity as long as decoherence doesn't corrupt the information too much. Often, interactions with the environment or spontaneous decoherence will pose strict limits on coherence times of qubit states.
For your wavefunction $\frac{1}{\sqrt{2}}\left(|0\rangle+|1\rangle\right)$, what you can do is initialise the qubit in the $|0\rangle$-state, and apply a control/pulse that maps this state to the desired state, such as the Hadamard gate.
Readout is performed through measuring the qubit in some basis. For photonic qubits, we measure the polarisation. For neutral atom qubits, we can perform a fluorescence readout.
