Which code in HTML will allow me to use an image as a background image for a table, but not so that it would be repeated several times vertically and horizontally (in case the table is several times bigger than the image), but in such way that the image height is stretched out to be equal to the height of the table, and its width is stretched out to be equal to the width of the table?
The CSS background options can't handle this reliably across browsers, so you need to put an <img> tag in the table and position it appropriately. As Petr Marek alluded to in the comments, you can do this with the CSS attributes z-index and position, but it's not elegant.
If you set position: relative on the table, you can set position: absolute on the <img> with top: 0; height: 100%; left: 0; width: 100%; to position and size the image, and set z-index: -1 to make it appear behind the other content.
Here's a working example on jsFiddle.
Although it works perfectly for me in Chrome, since you're putting content on top of an image I wouldn't be surprised if it caused some browsers to mess up text selection or something else.