I thought that promotional boxed sets were good candidates for the smallest box as well, as I own a 1630-1 Helicopter (which, incidentally, I got as a promotional gift for buying toothpaste at some point in the 90s IIRC), at 7 x 7 x 4.5 cm (220cm3/0.22 litres).
There are sets with similar dimensions as well, such as 5018-1 Gravity Games Promotional Set at 7 x 7.5 x 4 cm (210cm3/0.21 litres), 7912-1 Helicopter Promotional (Duracell), 5017-1 Hockey Promotional set. and some others.
I thought those were the smallest in volume, but after a deeper search on bricklink I came up with a thinner 2.5cm (281-1 1 x 2 and 3 x 2 Sloping Bricks, Red with 8.6 x 6.2 x 2.5 cm / 133cm3/0.133 litres):
And after delving in the sets with a low part cound, I came up with an even thinner 1.8cm box with 1222-2 1x1 round bricks, at 7.5 x 5.5 x 1.8 cm (74.25cm3/0.74 litres) (same box size as 1241-2, 1240-2, 1230-2 and possibly others):
Also of note are the 1:87-scale cars from the 60s, like e.g. 265-2 1:87 Karmann Ghia with Garage:
The "box" itself is a transparent hollow brick, specifically part number 721, 4x8 studs wide by 3 bricks high wrapped in a paper strip, which puts it at 3.2cm x 6.4cm x 3.04cm or 62.25cm3/0.062 litres. (Hey, you asked for the smallest box, not the smallest cardboard box!)
Apparently some of these 1:87-scale cars, e.g. 671-3 1:87 Vauxhall Victor Estate with Garage were sold with an actual cardboard box around them, with listed dimensions of 3.2 x 3.2 x 6.5cm (66.56cm3/0.066 litres):