The other two Solutions are correct,
you cannot use one field to initialize another. For this case, however, there is very a simple solution, define a constant for the default level value and initialize both based on that:
public class PublicPrivate : MonoBehaviour {
const float defaultLevel = 10f;
public float level = defaultLevel;
float health = defaultLevel * 10f;
You could even do better, by getting rid of as many in-line constants as possible in favor of a clearly named constant, and have:
public class PublicPrivate : MonoBehaviour {
const float defaultLevel = 10f;
public float level = defaultLevel;
const float levelToHealthFactor = 10f;
float health = defaultLevel * levelToHealthFactor;