Most package managers offer binary (already-compiled) packages. Most Linux/Unix software can't have its prefix changed *after* compilation. Everyone's user directory has a different path (thus, prefix). Thus, problems.
Options I've heard of & played with: GoboLinux's "Rootless" project is a system for installing from source in your home directory (on any distro). ZeroInstall I believe does source and binary (not sure how it manages binary) without root privileges. Some NixOS research has looked into rewriting the paths in compiled packages (though not changing its total-number-of-characters length).