It's called an improper prior. There's been some argument about their use but they seldom lead to problems. The posteriors usually has much better behavior at infinity and when they don't, that's the theory telling us that the information doesn't determine the solution to the problem.

The observation that an improper prior cannot be obtain as a posterior distribution is kind of trivial. It is meant to represent a total lack of information w.r.t. some parameter. As soon you have made an observation you have more information than that.