Any well-ordering of (say) the reals for which there is no largest element has the desired property.
For let $a$ be real, and let $A$ be the set of all $x$ such that $a\lt x$, where $\lt$ is a well-ordering. The set $A$ by assumption is non-empty, so has a smallest element $b$. Then the open "interval" $(a,b)$ is empty.
One can make many well-orderings of the reals such that there is no largest element. A natural well-ordering with this property is induced by a bijection between the reals and the smallest ordinal that has the same cardinality as the reals.
Any well-ordering of an infinite set, as long as there is no largest element, as the "empty interval" property. And the empty interval property always holds for all but the largest element, if there is one. In particular, sets with the property can be of any infinite cardinality.
The property that for any $a$ there is a $b\gt a$ such that the open interval $(a,b)$ is empty does not imply that the ordering is a well-order.
For call two reals equivalent if they differ by an integer. Let $E$ be the set of equivalence classes. Order $E$ arbitrarily. If $x$ and $y$ are real, put $x \lt y$ if the equivalence class of $x$ is less than the equivalence class of $y$, or if the equivalence classes are the same but $x$ is in the real number sense less than $y$. The ordering we get is not a well-order. However, it has the property that for any $a$ there is a unique $b$ such that $a\lt b$ and $(a,b)$ is empty.