Regions of outer space seem the natural best spots, outside a high-tech laboratory. That would ideally involve placing it in an area of near-pure vacuum, surrounded by high magnetic fields and maintained at extremely low temperatures.
The obvious general solution is to isolate the creature from its environment as much as possible. This means your creature likely will be unable to maintain this superposition for any significant amount of time. However, your creature will presumably be living in the natural world, not in a laboratory - and there are plenty of sources of perturbations to cause decoherence. Quantum computers can use error-correction codes and high magnetic fields, among other things, to account for and mitigate this decoherence. In quantum computers, decoherence happens on timescales of tens of minutes even in a best-case scenario. Minute interactions with the outside environment - heat, light, vibrations - can all destroy the superposition of the qubits. The loss of quantum coherence is a major experimental problem in, among other things, quantum computing. This is a process that involves the destruction of the superposition of a quantum system. Handwaving away some of the problems here, one basic one appears insurmountable: Decoherence.
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