You certainly can become a legal, Japanese citizen, with voting rights and the right to run for office and everything. The process to become a naturalized citizen is a long and complicated process, but it's certainly something you CAN do -- just look at Arudou Debito's site.
That said, permanent residency probably IS what the original poster is probably asking about, because becoming a citizen is a whole other ordeal that comes AFTER getting permanent residency.
Permanent residency information ->
Here
Becoming a real citizen ->
Here