For a somewhat serious answer --
Any of the Deep Southern states, pre-Civil Rights and the "Southern Strategy", could hold an all-Democratic legislature -- in fact, I wouldn't be surprised if they didn't at some point OTL.
More recently, Massachusetts has come pretty damned close -- there are so few Republicans that every member of the GOP caucus holds a "leadership position".
Still, this is a steep hill to climb because
a) The grand American tradition of gerrymandering tends to leave at least a few ultra-safe seats for both parties. A lock-out could probably only occur in a fairly small and homogeneous state -- Vermont, maybe, or a Dakota.
b) Most states are modeled after the federal Congress, with staggered terms for Senators ... so while you could knock out the House in one blowout year, it would take repeated punishments to lock a party out of both chambers.