Border doesn't support percentage... but it's still possible...

As others have pointed to CSS specification, percentages aren't supported on borders:

'border-top-width',
'border-right-width',
'border-bottom-width',
'border-left-width'
  Value:          <border-width> | inherit
  Initial:        medium
  Applies to:     all elements
  Inherited:      no
  Percentages:    N/A
  Media:          visual
  Computed value: absolute length; '0' if the border style is 'none' or 'hidden'

As you can see it says Percentages: N/A.

Non-scripted solution

You can simulate your percentage borders with a wrapper element where you would:

1. set wrapper element's background-color to your desired border colour
2. set wrapper element's padding in percentages (because they're supported)
3. set your elements background-color to white (or whatever it needs to be)

This would somehow simulate your percentage borders. Here's an example of an element with 25% width side borders that uses this technique.

HTML used in the example

.faux-borders {
    background-color: #f00;
    padding: 1px 25%; /* set padding to simulate border */
}
.content {
    background-color: #fff;
}

<div class="faux-borders">
    <div class="content">
        This is the element to have percentage borders.
    </div>
</div>

Issue: You have to be aware that this will be much more complicated when your element has some complex background applied to it... Especially if that background is inherited from ancestor DOM hierarchy. But if your UI is simple enough, you can do it this way.

Scripted solution

@BoltClock mentioned scripted solution where you can programmaticaly calculate border width according to element size.

This is such an example with extremely simple script using jQuery.

var el = $(".content");
var w = el.width() / 4 | 0; // calculate & trim decimals
el.css("border-width", "1px " + w + "px");

.content { border: 1px solid #f00; }

<div class="content">
    This is the element to have percentage borders.
</div>

But you have to be aware that you will have to adjust border width every time your container size changes (i.e. browser window resize). My first workaround with wrapper element seems much simpler because it will automatically adjust width in these situations.

The positive side of scripted solution is that it doesn't suffer from background problems mentioned in my previous non-scripted solution.

A pixel is the smallest unit value to render something with, but you can trick thickness with optical illusions by modifying colors (the eye can only see up to a certain resolution too).

Here is a test to prove this point:

div { border-color: blue; border-style: solid; margin: 2px; }

div.b1 { border-width: 1px; }
div.b2 { border-width: 0.1em; }
div.b3 { border-width: 0.01em; }
div.b4 { border-width: 1px; border-color: rgb(160,160,255); }

<div class="b1">Some text</div>
<div class="b2">Some text</div>
<div class="b3">Some text</div>
<div class="b4">Some text</div>

Output

Which gives the illusion that the last DIV has a smaller border width, because the blue border blends more with the white background.

Edit: Alternate solution

Alpha values may also be used to simulate the same effect, without the need to calculate and manipulate RGB values.

.container {
  border-style: solid;
  border-width: 1px;
  
  margin-bottom: 10px;
}

.border-100 { border-color: rgba(0,0,255,1); }
.border-75 { border-color: rgba(0,0,255,0.75); }
.border-50 { border-color: rgba(0,0,255,0.5); }
.border-25 { border-color: rgba(0,0,255,0.25); }

<div class="container border-100">Container 1 (alpha = 1)</div>
<div class="container border-75">Container 2 (alpha = 0.75)</div>
<div class="container border-50">Container 3 (alpha = 0.5)</div>
<div class="container border-25">Container 4 (alpha = 0.25)</div>