1) Binary files are 1s and 0s

2) Knitting has knit stitches and purl stitches

You could represent binary data in knitting, as a pattern of knits and purls…

You can knit Doom.

However, after crunching some more numbers:

The compressed Doom installer binary is 2.93 MB. Assuming you are using sock weight yarn, with 7 stitches per inch, results in knitted doom being…

3322 square feet

The idea of a “binary code” was originally developed in the textile industry in pretty much this exact form. Remember punch cards? Probably not! They were a precursor to the floppy disc, and were used to store information in the same sort of binary code that we still use:

Here’s Mary Jackson (c.late 1950s) at a computer. If you look closely in the yellow box, you’ll see a stack of blank punch cards that she will use to store her calculations.

This is what a card might look like once punched. Note that the written numbers on the card are for human reference, and not understood by the computer. 

But what does it have to do with textiles? Almost exactly what OP suggested. Now even though machine knitting is old as balls, I feel that there are few people outside of the industry or craft communities who have ever seen a knitting machine. 

Here’s a flatbed knitting machine (as opposed to a round or tube machine), which honestly looks pretty damn similar to the ones that were first invented in the sixteenth century, and here’s a nice little diagram explaining how it works:

But what if you don’t just want a plain stocking stitch sweater? What if you want a multi-color design, or lace, or the like? You can quite easily add in another color and integrate it into your design, but for, say, a consistent intarsia (two-color repeating pattern), human error is too likely. Plus, it takes too long for a knitter in an industrial setting. This is where the binary comes in!

Here’s an intarsia swatch I made in my knitwear class last year. As you can see, the front of the swatch is the inverse of the back. When knitting this, I put a punch card in the reader,

and as you can see, the holes (or 0′s) told the machine not to knit the ground color (1′s) and the machine was set up in such a way that the second color would come through when the first color was told not to knit.

Wires going through magnets.  This is how all of the important early digital computers stored information temporarily.  Each magnetic core could store a single bit - a 0 or a 1.  Here’s a picture of a variation of this, called rope core memory, from one NASA’s Apollo guidance computers:

You may think this looks incredibly handmade, and that’s because it is.  But these are also extreme close-ups.  Here’s the scale of the individual cores:

The only people who had the skills necessary to thread all of these cores precisely enough were textile and garment workers.  Little old ladies would literally thread the wires by hand.

With a Jacquard loom instead of painstakingly picking up every little thread by hand to weave in a pattern, which is what folks used to do for brocades in Ye Olde Times, this basically automated that. Essentially all you have to do to weave here is advance the punch cards and throw the shuttle. SO EASY. 

ALSO, it’s not just “little old ladies sewed the first spacesuits,” it’s “the women from the Playtex Corp were the only ones who could sew within the tolerances needed.” Yes, THAT Playtex Corp, the one who makes bras. Bra-makers sent us to the moon. 

And the cool thing with them was that they did it all WITHOUT PINS, WITHOUT SEAM RIPPING and in ONE TRY. You couldn’t use pins or re-sew seams because the spacesuits had to be airtight, so any additional holes in them were NO GOOD. They were also sewing to some STUPID tight tolerances-in our costume shop if you’re within an eighth of an inch of being on the line, you’re usually good. The Playtex ladies were working on tolerances of 1/32nd of an inch. 1/32nd. AND IN 21 LAYERS OF FABRIC.