How can I make a 300x300 pixels photo of myself without retouching or stretching my photo after taking pic?

Using Photoshop Elements, just crop to any square size. 5x5 is one of their presets. Next, resize the image. Hit [CTRL]+[ALT]+[I] (That's a letter "i") and type "300" in the space for pixel dimensions. If the proportions are constrained, it will change both and you'll be all done.

Alternatively, you can choose the crop tool and for the dimensions, set 300 px in one box and 300 px in the other. Whatever area you demarcate with the crop tool will result in a 300x300 pixel photo.

300x300 is next to nothing in size. Even the most cheap of cheap cameras can take that kind of photo. Most likely you will have to resize the photo using a photo editing program. I suggest using Paint.NET as it is free and rather easy to use. Also if you want it to exact 1:1 ratio, you will have to crop the image to ensure there is no stretching. If you really don't want to do this yourself. Send me the photo and Ill resize it and crop it for you as I work with photographs all the time.

300 Pixels

Depending on the size you want to print, you can use a quality 3.2MP camera to get good prints. I have a 5MP camera that makes great images, even printed on 8.5X11 glossy paper. Don't forget, the printer is a major factor in printed quality. Beyond that, anything above 5MP will work well. Most cameras today will be 8MP or above and still only cost around $100. Just stick with a known brand like Olympus, Nikon, Canon, Sony, Sanyo, Kodak, Fuji, etc. Ignore digital zoom, it will ruin you images, focus on optical zoom.

Take the picture of yourself. Open the picture in Photoshop. Click and select the Crop tool. The crop tool bar will show up on top. There are two boxes on the left side, one says width, the other says Height. If they are not in pixel (px), right click in the boxes and select pixels. Type in 300 for both. Now use your cursor to select the area you want to keep. The selected area will remain bright and the discarded area will turn dark. Click on the crop tool icon and select Crop on the popup window and you will crop the select area into 300x300 picture.