Have you looked at Puppy Linux ?
Precise Puppy is ~150Mb in size and you can use apt-get to install packages.
(it is based on the Ubuntu Precise repositories)
Otherwise there's tons of choice nowadays. Look for the ones that are Debian / Ubuntu / Knoppix based, you should be able to use apt-get with them.
As for the size, keep in mind distros like Damn Small Linux and Puppy can get so small only by stripping away as much functionality as they can, focusing on the bare minimum to run the system instead. These are the distros you'll find using using busybox and uClibc to save every byte they can. Feature rich environments like Gnome, KDE and even XFCE in this context push in the exact opposite direction, so you typically won't find them on a mini distribution.