See this thread: http://discussion.dreamhost.com/showthreaded.pl?Cat=&Board=forum_offtopic&Number=11085&Search=true&Forum=All_Forums&Words=postgresql&Match=Entire Phrase&Searchpage=0&Limit=25&Old=1year&Main=11085
In a nutshell, no. But feel free to enter it as a suggestion in the panel and convince lots of people to vote for it.
Technically, it would probably be possible to install and run it as your user (I think I caught a user doing that once), but we wouldn't allow that. On a dedicated machine, we'd happily install the package for you (I'm pretty sure we have for at least one customer), but wouldn't manage it or provide support for it.
And I'm sure folks here might be able to give you some ideas if you post specifics of what you're trying to do but can't... we run a pretty large system (DreamHost's backend, Dreambook) on MySQL, and haven't bumped into limitations too many times (we've been able to work around them when we have).