Raster images are what is produced by a camera, they're simply a recording of color at position. So a 300 DPI print is 300 color points per inch, which may or may not be necessary for what you need. The size of the image file required depends on the size of the print, and how many inches are you filling with 300 dots. (300 dots • width in inches = required width in px)
How are you planning to take a photograph that large anyway? You can't just scale it up; if you take, say, a 2000 by 2000 px image, plop it in Photoshop, and scale it to 2,000 by 2,000 px you haven't made the image any larger, just spread the same amount of detail out over more pixels. You've got a 6" wide print at 300DPI, or a 66" print at 30DPI.
Vector images are composed of mathematical definitions of shapes. Letters, for example, are vector: the shape of a Helvetica "R" can be printed at 10px wide or 10,000 px wide because it has a mathematical definition that is used to create a raster image (your screen itself is a raster image because it has a set, limited number of pixels) at any resolution.
However, just because vector images are mathematically defined and infinitely scalable, does not mean you can magically take a vector image with infinite detail. Detail and sharpness (around the edges of shapes) are different. In fact, if you used a computer algorithm to convert a photo into a vector image, you would lose detail, but the detail you retain would have crisp edges around each color area at any resolution.
In short: cameras take raster images, there's only so much size in a photo—there's no magic to make a really high quality print from a low resolution image. Take the resolution from your camera, find how large you can print at different DPI, and find what size and detail you want based on viewing distance and file size; 300 is not a magic number.