I owned a 70-300mm DO for a while. I was interested in it for similar reasons as the OP - small size for outings with the kids, taking the 24-105mm and the 70-300 DO instead of the 70-200/2.8L IS II. It's certainly great from a size perspective - about the same size as the 24-105, and while you can coax good IQ from the images, it does take more work in post than other lenses.
Here's an example:

EOS 7D, EF 70-300mm f/4.5-5.6
DO IS USM, 1/400 s, f/5.6, ISO 500
I owned the 70-300 DO for ~4 months before selling it. That was just before the 70-300mm f/4-5.6L IS came out, and I'd choose that over the DO lens now.
Piece of advice about the DO - if you are interested in one, buy it used. Unlike most high-end Canon lenses, where used prices are not much less than new, the 70-300 DO doesn't hold value - used copies sell on Craigslist for about $750. I bought one used, sold it for the same price I paid - basically a free 4-month rental. That's what I'd recommend if you want to try it out, but as I said, I'd prefer a 70-300L instead.
If it helps, I previously compared a bunch of lenses at a set of focal lengths, some primes and some zooms, including the 70-300 DO. You can see the lack of sharpness and especially contrast with the DO lens, but I added a set with sharpness/contrast boost, and it's ok.
Here are the links:
85mm
100mm
200mm
280mm