There are positional notations with bases that are not positive integers:
In systems with negative bases, both positive and negative numbers can be expressed with a single sequence of digits without any prepended sign for negative numbers.
Systems with complex bases extends this to being able to can express arbitrary complex numbers with a single sequence of digits.
These systems work well enough mathematically -- the reason they're mostly considered just intellectual curiosities is that it can be difficult for human readers to grasp numbers at a glance. They could be used for arithmetic internally in computers, but they still make life complex for the programmer, and in the majority of cases this outweighs the few theoretical nice points they have.
Systems with fractional or irrational bases can be imagined, but there we quickly begin to lose nice properties. The number of different digits we work with is by necessity always a natural number -- and since it cannot match an irrational base we either end up in a situation where there are significant gaps in the number line that we cannot represent (because there are too few digits), or we end up with numbers that have many different representations.
The problem of representations not being unique also arises (in a more tame form) with integral bases too -- in base ten we need to accept that $0.9999\ldots$ and $1.000\ldots$ represent the same real number, which occasionally confuses students a lot. But at least in an integral base, each number as at most two representations. In an non-integral basis, a number might have infinitely many different representations, and it would be very difficult to determine if two representations denote the same number -- or if they don't, then which of them is larger.