TLDR: Even if I’m off by 50%, it’s the mid 1970s for size. Weight? Like 2 of them mentioned the weight, don’t have the data, but double sided 5.25 inch floppies (1976) probably probably win on weight

Looking into this, it’s hard the dimensions of the large storage machines, they are often described vaguely or in a kinda anything-but-the-metric-system kinda way. So there is a lot of assumptions here that we just have to live with.

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That ends pretty close to your numbers, 700 MB to 330,000 pieces of paper, ~2121 bytes per page, ~1,060,606 bytes for 500 pages. I’ll use 500 per MB to math

Volume a piece of A4 paper is 0.3553 cubic inches. 500 pages is 177.65 cubic inches, therefore so is a MB stored in paper.

I won’t look much at weight since I could find it for most on them, but 500 sheets of paper weighs about 5.5lbs

Looking at an IBM 1311, available 1962 you get 12.6 MB in the size of washing machine. Average washing machine is 32,400 cubic inches. This gives us 2571 cubic inches per MB. So starting off ~14.5 times worse.

An IBM 2302 from 1965 stored 112 MB in 123,915.5 cubic inches, 1,106 cubic inches per MB.

The lowest capacity 8 inch floppies (80kb) available in 1971, 12.5 of them per MB, at a volume of 7.56 cubic inches a piece gives us 94.5 cubic inches per MB. This is just the floppies on there own, with a reader it would be at least large enough to not yet beat paper.

IBM 3340 (1973) and assuming it’s the basically the same size a 1311, and average washing machine, 32400 for 70 MB, 462 cubic inch per MB

Applying that same logic the later 3350 (1975) we get 32400 for 317 MB, 102 cubic inches per MB, which beats paper

Double sided 5.25 inch floppies (1976) 360 KB, 3.256 cubic inches each, I’ll round up to 3 per MB so 9.768 cubic inch per MB. Like the 8 inch floppies earlier, this doesn’t account for the size of the reader. I’d still say this is the point where we are beating paper for both size and weight.