I have a Casio Exilim. I take only 4 pictures and it says memory is full. It has no memory card in it. Is it possible that it only holds 4 pictures and I need a memory card?

Depending on the image resolution and quality the camera may hold anywhere from 2 to 69 pictures. Obviously smaller pictures take less space and thus you can fit more of them in. However, why buy a 6megapixel camera only to take pictures 2megapixels or smaller? The table on page 249 of the manual gives estimates for the various resolution and quality settings.

You're MUCH better off using a memory card. Even a 2GB memory card should let you take about 500 pictures, and the card can be reused after you download the pictures to your computer.