How to remove plastic shelf brackets that have broken off in the hole

A wooden shelf that is cantilevered 3 inches or so out from the end of a bracket will probably survive some load. But stack too much of a load on it (a pantry, canned food is heavy!) and even wood will crack. Particle board is very likely to fail of course as it has very little real strength.

A shorter bracket will often be designed for a smaller load. (Long brackets may be wider at the base, to support a larger load. They may be more robust to prevent buckling.) The issue is not just will it support the entire width of the board, but is the bracket adequate to support a greater load since a wide shelf will have more items stored on it. Those extra items stored are hanging a significant distance away from the wall.

Shelf construction

If the shelf has its front open edge positioned such that at each four feet the cable would go up to a separate roof truss than I think that you have a workable system. Make sure that your 2x3's under the outer shelf edges are placed to that the nominal "3" inch dimension is vertical.

Hardware

Note that the lowest cost eyebolts, known as the "bent type eye bolt" are generally not recommended for lift and continuous tension applications. For this application it may be suggested to search out the drop forged type eye bolts as shown on the right below. These will cost more but they are actually specified for "lift" type applications.

If you do use the bent type, orient them so that the split end of the eye is up, as show above. Definitely use through-bolts rather than any type of screw-in eye.

I would use 5/16" diameter eye bolts or larger.

Connection points

Put the upper end of the cable through a threaded eye bolt that goes all the way through the side of the rafter truss and is secured in the far side with a washer and nut. Avoid the use of a screw eye type fastener. The eye bolt should be attached to the upper rafter part of the truss instead of the lower cross tie. When drilling the hole for the eye bolt position it in the upper part of the rafter as opposed to the lower portion.

Put the bottom end of the cable through an eye bolt that is long enough to go through the shelf material and the 2x3 and secure on the bottom with a washer and a nut.

Note that the orientation of the cable clamps is important, and it is always advisable to use two cable clamps for securing the cable.

Cable sizing

The size of cable is really dependent on the amount of load that you expect for the shelf. I would think that 3/16" diameter cable would be adequate for this application. The chart below will help to explain why I recommended 3/16" (assuming you use the 1x19 type of "stainless steel aircraft cable").