Fixed position but relative to container

Yes, according to the specs, there is a way.

While I agree that Graeme Blackwood's should be the accepted answer, because it practically solves the issue, it should be noted that a fixed element can be positioned relatively to its container.

I noticed by accident that when applying

-webkit-transform: translateZ(0);

to the body, it made a fixed child relative to it (instead of the viewport). So my fixed elements left and top properties were now relative to the container.

So I did some research, and found that the issue was already been covered by Eric Meyer and even if it felt like a "trick", turns out that this is part of the specifications:

For elements whose layout is governed by the CSS box model, any value other than none for the transform results in the creation of both a stacking context and a containing block. The object acts as a containing block for fixed positioned descendants.

http://www.w3.org/TR/css3-transforms/

So, if you apply any transformation to a parent element, it will become the containing block.

But...

The problem is that the implementation seems buggy/creative, because the elements also stop behaving as fixed (even if this bit doesn't seem to be part of specification).

The same behavior will be found in Safari, Chrome and Firefox, but not in IE11 (where the fixed element will still remain fixed).

Another interesting (undocumented) thing is that when a fixed element is contained inside a transformed element, while its top and left properties will now be related to the container, respecting the box-sizing property, its scrolling context will extend over the border of the element, as if box-sizing was set to border-box. For some creative out there, this could possibly become a plaything :)

TEST

Actually this is possible and the accepted answer only deals with centralising, which is straightforward enough. Also you really don't need to use JavaScript.

This will let you deal with any scenario:

Set everything up as you would if you want to position: absolute inside a position: relative container, and then create a new fixed position div inside the div with position: absolute, but do not set its top and left properties. It will then be fixed wherever you want it, relative to the container.

For example:

/* Main site body */
.wrapper {
    width: 940px;
    margin: 0 auto;
    position: relative; /* Ensure absolute positioned child elements are relative to this*/
}

/* Absolute positioned wrapper for the element you want to fix position */
.fixed-wrapper {
    width: 220px;
    position: absolute;
    top: 0;
    left: -240px; /* Move this out to the left of the site body, leaving a 20px gutter */
}

/* The element you want to fix the position of */
.fixed {
    width: 220px;
    position: fixed;
    /* Do not set top / left! */
}

<div class="wrapper">
    <div class="fixed-wrapper">
        <div class="fixed">
            Content in here will be fixed position, but 240px to the left of the site body.
        </div>
    </div>
</div>

Sadly, I was hoping this thread might solve my issue with Android's WebKit rendering box-shadow blur pixels as margins on fixed position elements, but it seems it's a bug.
Anyway, I hope this helps!

Short answer: no. (It is now possible with CSS transform. See the edit below)

Long answer: The problem with using "fixed" positioning is that it takes the element out of flow. thus it can't be re-positioned relative to its parent because it's as if it didn't have one. If, however, the container is of a fixed, known width, you can use something like:

#fixedContainer {
  position: fixed;
  width: 600px;
  height: 200px;
  left: 50%;
  top: 0%;
  margin-left: -300px; /*half the width*/
}

http://jsfiddle.net/HFjU6/1/

Edit (03/2015):

This is outdated information. It is now possible to center content of an dynamic size (horizontally and vertically) with the help of the magic of CSS3 transform. The same principle applies, but instead of using margin to offset your container, you can use translateX(-50%). This doesn't work with the above margin trick because you don't know how much to offset it unless the width is fixed and you can't use relative values (like 50%) because it will be relative to the parent and not the element it's applied to. transform behaves differently. Its values are relative to the element they are applied to. Thus, 50% for transform means half the width of the element, while 50% for margin is half of the parent's width. This is an IE9+ solution

Using similar code to the above example, I recreated the same scenario using completely dynamic width and height:

.fixedContainer {
    background-color:#ddd;
    position: fixed;
    padding: 2em;
    left: 50%;
    top: 0%;
    transform: translateX(-50%);
}

If you want it to be centered, you can do that too:

.fixedContainer {
    background-color:#ddd;
    position: fixed;
    padding: 2em;
    left: 50%;
    top: 50%;
    transform: translate(-50%, -50%);
}

Demos:

jsFiddle: Centered horizontally only
jsFiddle: Centered both horizontally and vertically
Original credit goes to user aaronk6 for pointing it out to me in this answer