Not just "might", but is. It is a standard case of the discrete uniform distribution.

Other answers failed to give examples of a set with positive finite measure, which is what you really need. What we want is a set that has no upper bound, but isn't really weird or tricky—if it's too tricky, then the uniform continuous distribution doesn't make any sense—and an easy way to do this is take a union of a countable set of intervals.

For example, consider a set that contains (0,1/2), (1,1+1/2), (2,2+1/4), ... etc, so interval i has measure 2^-i. It should be obvious that there is no largest element, and if you sum the measures of all the sets you will find that the measure is 1. A big chunk of "doing mathematics" is having a library of techniques on tap to come up with weird objects like this to attack assumptions.

As a somewhat extreme case, the rationals (Q) are unbounded but their measure is 0, hence finite.

I suppose you could take [0,1] \cup Z, and put a uniform measure on that set.