It should weigh around 20 pounds. This can vary a bit though because the bags have varying levels of moisture can change the weight bag by bag.
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300 sq ft = 43,200 sq in [1 sq ft = 144 sq in]. 43,200 sq in x 6 in = 259,200 cu in of soil volume. The above assumes the garden is 300 sq-ft, but the question actually says 300 ft of fence length, not garden area. You would have alot of spare fence if your garden was only 300 sq-ft. So lets assume a rectangle with the dimensions of 100 ft x 50 ft (300 feet fence total legth), which gives an area of 5000 sq-ft. With the soil being 1/2 foot deep, the volume is calculated as 5000 sq-ft x 0.5 ft = 2500 cu-ft (or 92.6 cu-yards: 1 cu-yd = 27 cu-ft). However, as the dimensions of the rectangle change, so will the area, and so will the volume.
I presume the 6 mean 6 inches, so 1/2 ft. The volume of the garden is 4.5 cu ft. So you will need between 2 and 3 bags.
The answer depends on the thickness of the covering layer. That, in turn will depend on what the soil is for: lawn will require less depth than flower beds.
10 pounds
Approx 60 quarts.