Conjugate variables/operators are related by Fourier transform, that is, the (quantum) states of one observable are the Fourier transform of the other's and as such, only one of them can have a compact support (unless it's a zero function). This is known as the Uncertainty relation in Fourier transforms. Intuitively, it means the spread of a variable and its Fourier dual are inversely proportional, which physically translates into e.g. the position being localized (concentrated) and the momentum delocalized (spread out). For a proof approach see Qmechanic's answer.
Physically, all such types of variables/observables are incompatible (non-commuting $XP - PX \neq 0$, where $P \propto F^{-1} X F$ with $F: L^2(\mathbb{R}) \to L^2(\mathbb{R})$ ), as in they cannot be measured simultaneously to arbitrary precision. In other words, the uncertainties in the two variables are always bounded by the average of their commutator (even if you made the measurements separately on an ensemble of infinitely many identically prepared quantum systems). These uncertainties are an intrinsic property of any quantum state.