Explain Like I'm Five

I'm basically a tourist in quantum physics with no more than approximate understanding of several concepts, and I don't think I'll ever fully understand a field that took dozens of Nobel prize winners and multiple lifetimes to formulate

That’s everyone, honestly. Physics is big enough these days that I don’t think anyone could get all of it.

I always thought of "position" as simply a point in Euclidean space described by a vector, but I'm guessing that doesn't translate directly to quantum mechanics because the uncertainty principle gets introduced with having to account for momentum.

That very much still is the case (though it’s technically Minkowski space once you introduce special relativity); when you measure the position of an electron, you will get a single point as far as we can tell. It’s just that there is a range of locations you might see it in when you observe it.

Does that mean that two electrons can, at the instant they are observed, have no meaningful distance between them, only different momentum?

Hmm… yes?

I believe that two electrons ‘occupy the same space’ (down to some uncertainty) when they scatter off of each other. As stated above, they are point-like, though, so you would need infinite precision to make them properly overlap.

But there is a less finicky way to do it:

If you observe position (down to some accuracy), you can’t observe momentum (down to a related accuracy)—that is the core of the uncertainty principle. That being said, if you have perfect knowledge of their momentum, you will have no knowledge of their position, which will allow them to be ‘in the same place’ insofar as they both are everywhere all at once.

This can actually be done practically by cooling them down: if you constrain their temperature/energy/momentum, you can get them to ‘overlap’ through uncertainty. When this happens, they actually pair up, adding their one-half spins up to either 0 or 1. This integer spin makes the pair a boson and allows them to occupy the same states as other pairs (note that the electrons themselves cannot occupy each others’ states, but the pairs can, and these ‘Cooper pairs’ become the principle particles of interest). This lets them (the pairs) flow through each other without scattering, which is how superconductors work.

In classical statistical theory, manipulating a probabilistic state is equivalent to picking a single initial state with whatever probability, and then manipulating it. In quantum statistics it's provably not (at least if we're measuring particles with as much free will as we think); you need the whole thing for it to make sense. Two likely trajectories can interfere and cancel out, for example.

So, sure, a position is a vector. But we can only meaningfully talk about functions from a (measurable) set of vectors to their probability amplitude (which is like a probability, but complex). Or, in practice, the infinitesimal density of probability amplitude at that given point

The uncertainty principal is just one manifestation of that. And, like in the uncertainty principal, entanglement might not stay confined to just position if there's other parameters, so you really have to talk about functions on the whole state vector. I can't speak too much to quantum field theory, but the actual dynamics of basic quantum physics is about (very "basic") functions on those functions.