Well, in the coin flip case, the thing you care about learning about isn't the value in {Heads, Tails} of a coin flip, but the value in [0,1] of the underlying probability that the coin comes up heads. We can then put an improper prior on that underlying probability, with the idea that after a single coin flip, we update it to a proper prior.

Similarly, you could define here a family of distributions of string lengths, and have a prior (improper or otherwise) about which distribution in the family you're working with. For example, you could assume that the length of a string is distributed as a Geometric(p) variable for some unknown parameter p, and then sampling a single string gives you some evidence about what p might be.

Having an improper prior on the length of a single string, on the other hand, only makes sense if you expect to gain (and update on) partial evidence about the length of that string.