Well you could use Bundle::DBD::CSV or DBD::Sprite emulate SQL queries over flat files. DBD::RAM is probably faster, because it does it in memory. The former are probably too slow if you have more than a few dozen records.

DBD::Trini looks great, but is in early stages of it's implementation.

Personaly, I would look at DBD::DBM || DBD::MLDBM as your best bet.

I would also consider changing hosters. Cheap hosting that provides mysql is pretty easy to get.

Update:(Thanks to dragonchild) DBD::Trini, not DBD::Trinni.