Scooters in Barcelona
I have never actually done this, as I believe that my feet and the Barcelona metro service are the best option for touring Barcelona. However, renting scooters is yes very common, and there are plenty of sites out there for scooter rentals. A quick google search revealed these.
I clicked on a random one, set bogus dates (from 01/08/2015 to 01/09/2015), picked the scooter I thought is newest and got a total of 345,63 €. (TAXES INCLUDED). Add 30Euros for extra insurance lowering the franchise to 300Euros (15Euros per week) and you are looking at less than 400Euros for a 31 day rental.
Here is a screenshot:
It could probably be worth booking in advance, as I can only assume that prices levitate during high season, and as scooter availability drops. Also, I would stick with large companies (the equivalent of Hertz/Europcar so to speak) to make sure you are getting proper service and true legal policies behind the rental. Some guy on the street with an extra set of wheels to rent might be cheaper, but you never know how things might work out if you happened to crash the scooter. In these cases it is better, I think, to have a proper rental contract, with guarantees.
Larger Wheels, Man
On a different, technical, note I would advise you to go for a scooter with large rims (in terms of diameter) as these provide more comfort when driving in the city. Smaller rims allow for manoeuvrability at low speeds and fast cornering, whereas larger rims roll easier over obstacles and are more stable at high speeds. I assume you are no Marc Marquez so you don't really care about the speed issue. However riding over potholes, and boy are there potholes-covered roads in Europe, with larger wheels is way more comfortable than with smaller wheels. For more info see here.
There are other technical considerations you should keep in mind such as: are you going to use it only in the city? What's the average distance you plan on riding every day? I'll leave those for another question.