As Mark said, 9V Alkaline batteries aren't the best choice for continuous high current discharge. See the following for data on the EverReady Energizer and Duracell (the Duracell data is better):
http://www.data.energizer.com/PDFs/522.pdfhttp://www.duracell.com/media/en-US/pdf/gtcl/...uct_Data_Sheet/NA_DATASHEETS/MN1604_US_CT.pdfBut, if the high current is intermittent, then perhaps you could get away with it, but the capacity will still be reduced quite a bit. Another possibility would be 3 9V batts in parallel.
Lithium would be a better choice but at 300ma, still a reach (unless intermittent):
http://www.duracell.com/media/en-US/pdf/gtcl/Product_Data_Sheet/NA_DATASHEETS/DL1604-9V Ultra Lithium Dioxide.pdf
Again, 2 or 3 in parallel might do the trick.
Now, when you say 300ma and 5V, do you mean that you plan to use a DC to DC converter? If so, with, say 85% efficiency, you'll be drawing around 200ma from the 9V battery, still a bit high, but better. If you mean, draining the battery down to 5V, then check out the discharge curves in the data sheet links, above (you'll have to extrapolate, as none spec at 300ma).
