I can store probably around 33 million files without seriously affecting lookup time (256 * 256 * 512 - I'm allowing a 50% margin), but backing up or transferring these files would be a major pain

Gee, I don't know, it would seem that making incremental backups would be easier than with a normal database.

But what about database searches? Seems to me that there I'd also have to have a marged copy of the text there in a processed (punctuation and extra spaces removed) format, and just searching the text as it resides in the database won't be much help.

But you definitely should be thinking about about indexed searching, where you make a reversed index of important keywords and a list of every post where it can be found.

Will storing the posts in the database instead of as individual files take up a lot more space

As a general rule, databases are wasteful with space. They all seem to be designed as if disk space was free: if your disk isn't large enough to hold the data, you can just throw more hard disks at it. So no, I don't expect databases to be space efficient, or even recycle wasted disk space by itself. Whatever you do, your home grown solution will likely be a lot more compact.

But be prepared to put a lot of effort in it. To make a decent database-like system from scratch is a lot of hard work, requiring a lot of nifty home-grown solutions, many of those which are simply part of a standard database system.