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It's possible but it'd be one heck of a long day. A quick Google search brings up a few companies running day trips to Petra from Sharm, costing around $200 each and taking a whopping 20hrs. 
There is a catamaran from Sharm to Aqaba three times a day (the sail takes an hour) so it won't be impossible to do the journey independently, but maybe a bit of a logistical nightmare if you're relying on public transport to get from port to Petra and back again in time for the last boat back. 
Cant help you re car hire, but maybe others can help. Failing that Trip Advisor forums are teeming with that kind of info. 
Hope this helps 

Thanks very much for that, great help. Will advise friend, who is here, to search Trip Advisor. :-)

Nothing constructive to say, but so beside myself with excitement to be going there in October I wanted to join in. I've booked a week, but know it won't be long enough. Can't imagine going for just a day!
My advice would be to cancel Sharm and go to Jordan.

We hired a car in Aqaba and did this trip with an overnight stop in Madaba as the distance is so great, and of course Petra is not to be rushed. One day would be crazy.....doubt they could possibly get back to Aqaba in time for the last boat to Sharm either. 

Arranging hire cars in Jordan is a little different but do-able through one of the hotel bellmen in Aqaba.

Thank you all :-)

Petra and the Dead Sea would be doable in a day starting in Jordan but, as the others have said, from Sharm it will be one hell of a day, especially if you want to see anything meaningful in Petra or climb up to the monastery or high place of sacrifice.  Overnight would be more feasible but even then you will only scratch the surface of Petra.