Certainly it would have to be something subatomic. Would it be the diameter of an electron? Does an electron even have a measurable diameter, what with the Heisenberg principle and all? Or the diameter of a quark? Or would it be the distance between the lowest and next-lowest electronic orbits? Or the diameter of a hydrogen nucleus? Or what?
It’s called the Planck Length .
It’s Very Very Small.
Wow, my penis is approximately 111250000000000000000000000000000000000 Plank Lengths long. I feel like John Holmes 
As well you should. Because if your math is correct, and my checking of it is also correct, we’re talking just over a mile!
Perhaps a decimal point got misplaced?
Well, he did say approximately…
Can we measure a plank length? I did not think that we could get anywhere near measuring things that accurately.
It’s a Planck length. Asking how long a plank is, is like asking how long a piece of string is 
(Though if you have a specific plank in mind, you can measure it with a tape measure).
Can we measure strings shorter than a Planck Length? I’m a frayed knot.
If the question is a practical one, rather than theoretical, the LIGO gravitational wave detector measures changes in the distance between two masses down to 10[sup]-18[/sup] meters (for comparison, that’s about 10[sup]17[/sup] Planck lengths). I think this is smaller than the measured bounds on the size of the electron, though I’m having a hard time finding a number for that (electrons are believed to actually be true point particles, but things like that can’t generally be experimentally confirmed: All we can say is “it’s smaller than so-and-so”).
A slight hijack but i have tried to read up on Planck units and do not understand them. Are they just the smallest units possible?
It is the smallest measurable unit of length. Beyond that, you get lost in the “quantum fog” no matter what methods you use to measure a particular distance.
There’s another current thread on this, but to sum up: We have a pretty good theory of gravity, General Relativity, which has passed all of the tests we’ve put it to. We also have a pretty good theory of quantum mechanics, which has also passed all of the tests we’ve put it to. But quantum mechanics as we know it and general relativity as we know it are fundamentally incompatible. The only reason neither of them has failed any tests is that it’s very hard to find a situation where they’re both relevant. If one were to have such a situation, we can confidently say that we have no clue what would happen in that situation, since we don’t have a theory of quantum gravity.
Well, if you had some phenomenon occuring at the Planck scale, both theories would be relevant to that phenomenon. They might both be relevant on some more accessible scale, but we’re certain they’d be relevant at the Planck scales. So, we can confidently say that we don’t know what would go on at scales comparable to the Planck scales, or at scales more extreme.
10[sup]-18[/sup] m is actually also the current upper limit on the electron radius.
Your correct, my penis is only 11125000000000000000000000000000000 Planck Lengths long 
Sure, it looks big, but that’s comparing it to a really really small unit.
Depending on temperature, you might still be over or under by one zero on the end.
OK, I found several different numbers, including that one, in my search, but couldn’t tell which one was correct. I’ll take CERN’s word as authoritative, though. I highly doubt there’s any other subatomic particle for which better bounds are known (quarks are also believed to be point particles, but they’re much harder to study than electrons), so it looks like LIGO and the electron size bound are running neck and neck for the record.
The smallest unit of length that has a name (in the traditional metric prefix nomenclature) is the yoctometer
= 1 x 10[sup]-24[/sup] meters
If someone went to the trouble of actually assigning a name to this quantity, perhaps they thought there was some meaningful measurement for it?
Nah, you can do better than that with a yoctoångstrom.
But just because the prefix exists, and the meter (or ångstrom) exists, doesn’t mean they were specifically meant to go together. A yoctogram, for instance, is a perfectly usable mass unit (about .6 amu), so that would be enough justification to create the prefix.
Yes, I think that electron limit is currently the tightest such bound on a particle radius - though it wouldn’t surprise me if there have been times over the years, or in future, when that for the muon held the record.
By comparison, the best limits on quark radii are, I believe, roughly 10[sup]-16[/sup] m.
That it has a rival in this context shouldn’t, of course, detract from just how gobsmacking LIGO’s precision is.