Can a Nikon d5000 shoot images at 300 dpi?

Can a Nikon d5000 shoot images at 300 dpi?

Yes.

Setting the image quality to fine and the image size to large will produce an image file with the equivalent of 300 dpi (dots per inch).

No camera can shoot at any amount of DPI. Why? Because DPI stands for dots per inch. DPI is a measurement used to describe the resolution of a print and not the camera's resolution which is measured in PPI or pixels per inch.

The camera shoots a certain number of pixels, it knows nothing of inches. It's up to the display device or printer to interpret the image in terms of DPI. The maximum image size is 4,288 x 2,848. Printed at 300 DPI it would make a photo 14.3 inches by 9.5 inches.

There's lots of confusion about dpi and digital cameras. As others stated, dpi has no meaning with digital cameras. It has meaning when printing and scanning.

To confuse matters, so called photo contest sites ask for digital images with a certain dpi setting - utter nonsense!

Even more confusing is the EXIF header in jpeg files have a dpi setting. Various camera manufacturers put in random numbers like 72 dpi (a throwback to the 1980s MAC display), 300 dpi (commonly used for printing images), and sometimes some other number. The dpi number in an EXIF header does not affect image resolution or quality. It's just a number with no meaning.

As others noted, the only thing you're interested in, when it comes to resolution, is the number of pixels that make up your image.

AFAIK, digital cameras shoot at 72DPI. You can easily change that setting in the computer with a capable graphics program. If you have none, try Irfanview (it's free).

Your question is meaningless.