the room is 16x24 and the ceiling joist are on 24 inch centers..My question is do I need a support beam in the middle of the room to hold the 16 foot span to keep it from swaying or will the 2x4's be enough?

While Walks on Water or Aminator are the best people to comment on this, I have to point out that most building codes require bigger lumber than 2x4s for residential ceiling joists (and 16" oc rather than 24"). I know that you are talking about a storage building, but, in my opinion, it certainly won't hurt to overbuild in this case.

Flat 'ceiling' or roof will most likely need to be supported with a beam running down the center. This depends on your load:

http://www.mcvicker.com/resguide/page013c.htm give span limits/and estimated loads depending on various dimensions of lumber (2x4, 2x6, etc) and the types of wood use (Fir 1 and 2)

You will note that the limit for roof rafters are 2 x 6, this is assuming static load (roof members, underlay, roofing material) and takes into account average loading from things like snow.

I am under certain if you are talking about a ridge beam (the long beam at the peak of a roof) or if you are planning on running a crawl/space or 'loft'.

In any case you should aim for 2x6's to be on the safe side.