Interesting question and there are scientists thinking about things like this. I am not a physicist, so my opinion doesn't count for anything, but here's what I think. I want to try answering it anyway because I learn a lot from thinking out loud and then letting others correct my thinking. So here it goes.

Given enough time, it might be possible to scientists to invent something like this. We will eventually know more about producing and storing this kind of energy. Gamma energy is photon energy and while photons do have mass, it is very small. However, because e=mcc that energy could be converted to mass. But this is not what you are proposing, you are working it the other way, converting a mass into high energy and then using that energy to propel the ship. If I understand you right you are proposing to produce and store the energy off ship and then loading the ship with the stored energy.

I think there are two main problems:

1) From what I've seen, people who want to produce positron engines have proposed using the gamma radiation to head a fuel source to combustion, not using the gamma energy as the primary thrust. Low mass photons won't do much to overcome the inertia of the ship. I imagine that if it did work then what you would have is a giant high energy beam that would annihilate anything in it's path for a very long way.
2) By straws, I suppose you might mean nano tubes. This seems like a nice idea, but you would need more than millions. Even billions of photons produce very little mass. So it seems to me that the storage means weight and weight is the enemy here. Whatever energy would be produced would have to be immediately used in order to make the system efficient enough to be practical.

I think that because of the energy and weight requirements of your proposal, that a better solution would be to produce the gamma radiation off ship and beam it to the ship via a laser. This might keep the ship light enough to make it feasible.

Nevertheless, I think scientists should work on something like this since, even if it did not work out as planned, we could learn a lot a basic science by working on it.
Source(s):
http://www.universetoday.com/2005/06/30/positron-drive-fill-er-up-for-pluto...
http://tinyurl.com/ydbyxac

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