I present to you a light giant (you asked for OEW).
One of the major reasons to have 4+ engines* is if the engines are weak, and the plane is big. Therefore one of the lightest will be an early 20th century giant. Jets are out of the question since they are inherently heavier and most (if not all) are all-metal.
From the Wikipedia Category:Four-engined_aircraft, the only two planes under push configuration are both experimental. That leaves us with the push/pull configuration if we are after light planes (common nacelle, fewer stress points, etc.).
The lightest (but not smallest) I found and hence nominate is the Gotha WD.27 at an empty weight of 4,500 kg and powered by four 160 h.p. Mercedes D III's driving 4 propellers (each two in push/pull). It weighs as much as 5 Cessna 182's, after all it has 4 (old and heavy) engines.
I doubt we can find any 4-engined fuel powered airplane under 4 tonnes (c. 1913 it was possible, see comment by PK). And piston engines aren't easily miniaturized (dang it, see xxavier's answer).
Just look at it, it's nothing but a giant wing with 4 engines and two floats. Reportedly 3 were built for the military for sea patrol. There is not much information about it online. An aircraft encyclopedia I have doesn't say much about it either, only confirms 3 were built. (The smaller cousin, the WD.22, was a prototype at 3,800 kg.)
* I didn't say 3+ because 3 can be a workaround for the era pre-ETOPS.
There's the beautiful Dornier Do 26, but unfortunately it's over 10 tonnes.