After measuring momentum, it seems like the particle's position could be literally anywhere? Once measuring momentum, the wavefunction "collapses" into something that looks like this

If you were to then measure the position, couldn't it be literally anywhere? What am I missing? Is it even possible to measure momentum perfectly?
 A: We call a wavefunction that has a precisely defined momentum a momentum eigenstate. For a free particle the momentum eigenstates are infinite plane waves like the one you show in your graph:
$$ \psi = e^{i(\mathbf p\cdot\mathbf x - \omega t)} $$
And as you say in your question for this eigenstate the position of the particle is completely undefined, or put another way $\Delta x = \infty$.
But how are you ever going to make a measurement that results in an infinite plane wave? What possible physical process could achieve this? Any measurement necessarily takes place within some finite region so the best you can achieve is to end up with a wavepacket that is about the size of your system:

where $x$ is some length scale determined by how you did the measurement. The resulting wavefunction will be:
$$ \psi = \mathcal F(\mathbf x,t) e^{i(\mathbf p\cdot\mathbf x - \omega t)} $$
where $\mathcal F(\mathbf x,t)$ is the envelope function. However this wavepacket no longer has a precisely defined momentum because it is not an infinite plane wave so it isn't a momentum eigenstate. In fact the momentum spread will be roughly given by:
$$ \Delta p = \frac{\hbar}{2x} $$
i.e. just the uncertainty principle. So as a consequence of your measuring apparatus having a limited extent in space you can only measure the momentum to a limited precision. Your measurement does not cause the wavefunction to collapse to a momentum eigenfunction and the resulting particle cannot be anywhere in space.
A: Each observable corresponds to a mathematical operator in Hilbert Space. There are pairs of observables which are called conjugate variables, these cannot both be known accurately at the same time. The measurement of one immediately makes the measurment of the other impossible. Position and momentum are such a pair. It is possible to measure one to a lesser precision, and then measure the corresponding conjugate variable in a similar manner. The relationship between the accuracy of the two measurments is given by the Heisenburg uncertainty.
A: Real measuring devices all have a granularity. Your experiment will never tell you that a particle has momentum exactly $4.03752\,\mathrm{MeV}/c$; it will tell you (assuming it is very precises indeed) that it has momentum $(4.037 \pm .014)\,\mathrm{MeV}/c$ which is completely compatible with the particle still being found withing the apparatus at the end of the measurement.
A: "We have measured sample's momentum with absolute accuracy, so nobody knows where it is now!"
You are right and not missing anything.
