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It’s easy to use a uniform prior in a suitably huge interval - say from minus one gazillion to plus one gazillion - and you can look at the limit when the endpoints go to minus/plus infinity if you really have doubts about whether the interval was huge enough for your problem to fit comfortably within it.

If you don’t think that this is possible that says more about you than about the shortcomings of Bayesian statistics.

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What would you do if you see an observation outside of the interval?

Use a larger interval?

As I said, you can define the improper uniform prior solution as the limit of a sequence of solutions corresponding to a sequence of increasingly wider intervals with endpoints that go to +/-infinity.

(And as I said, you can start with a suitably huge region. Say that you want to determine the position of something and use a uniform prior that extends to a distance of 10^27m - a perfectly bounded prior from a mathematical point of view that covers the whole observable universe. If you observe something outside it, it’s not with the prior that you have a problem.)

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