Quibble quibble. First, there is no guarantee at all that any particular sequence is
in the digits of π, unless you've already calculated that far and found it. A common misconception, but see is pi normal?
for the current state of play. So we can't be sure we'll find the Gödel number
of the Britannica
in that particular representation.
So let's store it as the data of a quantum computer. Toothpicks are made of wood: what's wood made out of? Carbon, oxygen, hydrogen, a few other elements. And how much does one weigh? Let's use a big chunky toothpick that contains a whopping 2 g of atoms the average size of carbon atoms: that's one sixth of a mole, and therefore contains 1/6 x Avogadro's number of atoms. That'll do for a round number: 1023 atoms.
Call the content of the Britannica a round thousand million bits. Then each atom has to store 1014 bits, or about 250. A much more manageable number.* All we need now is a bit of quantum superposition
Read footnote, not this next bit...
Hmmm... tricky. A carbon atom's got about twelve nucleons and twelve electrons, oxygen a few more, but on average about 25 particles per atom. Each of these might have a few discrete states it can be in, like spin, but I foresee falling short if we rely on those. Let's instead just consider the six to eight shell electrons on each atom... say 8 because I can divide it into 250.
So we need to excite every electron in the toothpick simultaneously into the first 247 or so energy levels above their ground state.
* Hey, tdent points out an even easier way, which involves not getting the maths wrong. 1023 atoms storing 109 bits is a whopping 1014 atoms per bit. So we can pick and choose which of them get used in the superposition, and maybe even build a detector out of the rest.
As tdent said, the paper Britannica itself is not many orders of magnitude larger than a toothpick. Let's say it's 10 000 times bigger in round numbers, 20 kg as against 2 g. This prompts the thought of whether we could actually keep the text on the paper copy: how much heavier is a sheet of printed paper compared to the same blank? A ten-thousandth more? Throw away the heavy covers. Keep just a representative portion (say a central line preserving the shape) of the printed letters, attached to a continuous carbon nanotube: if that's thin enough it should no more than double the mass. Then wind this around the toothpick.
Nanotubes, did I say? You could probably do it in DNA and set up your own miniature printing press.