C# Ping - Checking availability of hosts
It doesn't make sense to have the CancellationTokenSource private like you have now. Who is going to be able to cancel the operation? The user of your class can't access the source or provide their own token. The class doesn't offer any API to cancel.

Instead, you should alter the private bool IsOneOfTheHostsAvailable() method to accept a CancellationToken as input. You can then let the user care about creating a source and possibly cancelling the operation. Or, if you don't feel like offering the option to the user, you should just do away with the cancellation token as a whole. Right now it isn't doing anything.