I would like to be able to install packages by using a package manager (Yum is available on my server), but I don't have root access. I don't technically need root access, I just want the packages to be installed in my home directory.
Is there any way I can do this?
Not using yum. You can use yumdownloader though, and then put the packages through rpm2cpio ... | cpio -id in order to extract their contents, and then place the files in the appropriate locations.
As Ignacio points out, you can use yumdownloader.
But for the second step
rpm -ivh --relocate /usr=/home/username/usr --relocate /etc=/home/username/etc
or
rpm -ivh --relocate /=/home/username/progname
is probably easier.
If the RPM has any scripts that assume they are running as root, you might also need to add the --noscripts option.
Plus you can add the --aid option to resolve dependencies.
Yet another useful options in this case is --root to avoid rpm to write to /var path (since you're not root), so the command will be something like:
rpm -ivh --relocate /=/home/username/progname --root /home/username/progname
