does miata have a safety stop when oil switch fails

No. The safety stop is supposed to be the driver. The oil gauge doesn't read level, it reads pressure. If it read zero, the oil pump died. Does the engine turn over when the key is moved to Start position or do you just hear a clicking sound?

Yes, when I turn the key to the start position the engine turns over. So now what do I do? Just replace the oil pump?

Yes, When I turn the key to the start position the engine turns over. Does this mean all I need to do is replace the oil pump?

Maybe. You need to determine what the problem is before throwing parts at it. Rent or borrow an oil pressure gauge. You can usually do this at Advanced, Autozone, Pep Boys, etc. Remove the oil pressure switch and install the fitting or sending unit for the gauge. Crank the engine over to see how much pressure, if any, is built up. If none, then you need to replace the oil pump. If you get at least 15PSI, you need to replace the oil pressure switch.

Once that is taken care, you need to determine why it isn't starting. Make sure there is fuel and spark.

What you need to do here is verify that this elements of combustion: compression,fuel and spark are all in order