There are options for installing software without root access.
First of all, using the sources (if available) you can build and use most packages as a non-root user -- but this "from sources" way forces you to deal with dependencies, as well.
Starting from that, pkgsrc makes this easier, by offering a large number of, say, recipes to build packages and their dependencies (so-called "ports").
It supports non-root operation, see this FAQ entry. There is a guide on how to use pkgsrc here.
Regarding the packages you're looking for, they have "ports" in pkgsrc:
net/skype21. (Where Chromium, the Open Source project related to Google Chrome, is available as a work-in-progress package only, i.e. via
wip.sf.net, a repository of additional ports to pkgsrc.
Also, there is ZeroInstall, which as far as I know provides binary packages that you can install in some way not requiring root permissions. (Of course you need to install ZeroInstall first, which you can -- if your sysadmin doesn't install the
zeroinstall-injector Debian package for you -- install from source, too, as explained here.) I'm not sure if this way is less involved, though. (From a quick look, they seem to have Chromium and Skype packages, but the first one looks kind of old.)
The bottom line: it is possible, but involves some work and things to learn (which can be considered a good thing, while time-consuming). The easiest way probably is calling your sysadmin to install the Debian packages you want to use.