Yes. It s not too tricky, and the following will work in all mainstream browsers (including IE 6, and indeed 5):

(Updated 7 September 2012 after Jukka Korpela s comment pointing out that the previous version didn t work in IE 9 standards mode)

Demo: http://jsfiddle.net/timdown/hGkGp/749/

Code:

function selectElementContents(el) {
	var body = document.body, range, sel;
	if (document.createRange && window.getSelection) {
		range = document.createRange();
		sel = window.getSelection();
		sel.removeAllRanges();
		try {
			range.selectNodeContents(el);
			sel.addRange(range);
		} catch (e) {
			range.selectNode(el);
			sel.addRange(range);
		}
	} else if (body.createTextRange) {
		range = body.createTextRange();
		range.moveToElementText(el);
		range.select();
	}
}

<table id="tableId" border="1">
	<thead>
		<tr><th>Heading 1</th><th>Heading 2</th></tr>
	</thead>
	<tbody>
		<tr><td>cell 1</td><td>cell 2</td></tr>
	</tbody>
</table>

<input type="button" value="select table" onclick="selectElementContents( document.getElementById( tableId ) );">

Just to make the code proposed by Tim Down more complete, allowing de selected content to be automatically copied to clipboard:

<script type="text/javascript">
    function selectElementContents(el) {
        var body = document.body, range, sel;
        if (document.createRange && window.getSelection) {
            range = document.createRange();
            sel = window.getSelection();
            sel.removeAllRanges();
            try {
                range.selectNodeContents(el);
                sel.addRange(range);
            } catch (e) {
                range.selectNode(el);
                sel.addRange(range);
            }
            document.execCommand("copy");

        } else if (body.createTextRange) {
            range = body.createTextRange();
            range.moveToElementText(el);
            range.select();
            range.execCommand("Copy");
        }
    }
</script>

<table id="tableId">
    <thead>
        <tr><th>Heading</th><th>Heading</th></tr>
    </thead>
    <tbody>
        <tr><td>cell</td><td>cell</td></tr>
    </tbody>
</table>

<input type="button" value="select table"
   onclick="selectElementContents( document.getElementById( tableId ) );">

I got it to work in IE9 finally by using the following script

NOTE: It doesn t work on html tables. It HAS to be a DIV. So just put a wrapper DIV around the table you need to select!

First I changed the HTML button code a bit:

<input type="button" value="Mark table"    onclick="SelectContent( table1 );">  

Then changed the javascript to:

function SelectContent (el) {


var elemToSelect = document.getElementById (el);

        if (window.getSelection) {  // all browsers, except IE before version 9
            var selection = window.getSelection ();
            var rangeToSelect = document.createRange ();
            rangeToSelect.selectNodeContents (elemToSelect);

            selection.removeAllRanges ();
            selection.addRange (rangeToSelect);



        }

    else       // Internet Explorer before version 9
          if (document.body.createTextRange) {    // Internet Explorer
                var rangeToSelect = document.body.createTextRange ();
                rangeToSelect.moveToElementText (elemToSelect);
                rangeToSelect.select ();

        }

  else if (document.createRange && window.getSelection) {         
          range = document.createRange();             
          range.selectNodeContents(el);             
    sel = window.getSelection();     
                  sel.removeAllRanges();             
    sel.addRange(range);              
 }  
}

Here is what I have used. It also make the copy command so all you have to do is use the paste command in the document you want to place it into. Basically you wrap the content you want to copy in a div, grab it using innerHTML and copy it to the clipboard. I have not tested it on all browsers but it works in Chrome, Safari, Firefox.

<div id="copycontent">
<table>
</table>
</div>
<input type="button" value="Mark table"    onclick="SelectContent( copycontent );">

<script type="text/javascript">
function SelectContent (el) {    
  var aux = document.createElement("div");
  aux.setAttribute("contentEditable", true);
  aux.innerHTML = document.getElementById("main").innerHTML;
  aux.setAttribute("onfocus", "document.execCommand( selectAll ,false,null)"); 
  document.body.appendChild(aux);
  aux.focus();
  document.execCommand("copy");
  document.body.removeChild(aux);
}
</script>