Flooded I'd guess. When you last started the engine did you run it only for a few seconds, ie. just to move it? If you did then it's almost certainly flooded. These do have 'Clear Flood Mode'. Crank the engine with your foot hard down on the throttle. No fuel will be injected if you do this and you may be able to clear the flood. Then crank the engine with no throttle to try to start it. If this fails take the spark plugs out and clean them and try again.

When you say the engine attempts to turn over do you mean it does crank or it doesn't? So to start again does the engine crank over well when you turn the key to start it? Or does it crank slowly as if the battery was flat? RX8s have rotary engines which are very simple in design and there's not that much actually to go wrong with them, but they are quite prone to flooding.

Assuming the engine cranks over well is it trying to fire? If it is I'd suspect it's flooded. If not do you have a good spark at the spark plugs? There are two plugs per rotor (a leading plug and a trailing plug), and each plug has its own ignition coil. The plugs are not that easy to access, I accept that, but you may need to remove one to check its condition and to see if it's wet with fuel.

If the engine does not crank over well try recharging the battery or use a boost pack or jump leads.