Illustrator does not have any way of doing this directly implemented. However as @Scott mentioned there are third-party plugins that do implement this.
As a simple workaround when you want to change the length of just one Handle you can mark the current location of the handle relative to the anchor using either the line tool or two guides pulled out from the ruler.
For the workarounds ensure you have Smart Guides and Point Snapping enabled.
Line Method: Select the Line Tool and drag from the Handle you want to maintain to its associated Anchor. Then adjust the opposite Handle as you desire. When finished drag the first Handle back to its original position at the end of the Line you made earlier. With Point Snapping enabled the handle should snap to the Anchor at the end of the Line.
The advantage of this method is that if you need to move the Anchor as you make edits to the second Handle, you can easily move the Line with the Anchor and put the first Handle in the same spot relative to the Anchor after the move.
Guide Method: Drag a vertical or horizontal Guide out from the rulers at the edge of the screen and snap it to the Handle. Then re-select the Anchor so the Handle is again visible and drag a Guide out from the ruler perpendicular to the first, and snap it to the handle. After editing the other Handle, use the intersection of the two Guides to snap the first Handle back to its original position.
If you want to change the length of both handles, independent of each other or not, but maintain the same angle then use the Line Method. But after marking a Handle, deselect everything and click the Anchor while the Line Tool is still active. In the dialog that opens change the length of the Line but not the angle. Now when you drag the Handle it should snap to the Line and you can set the length of the Handle freely along that line.
In newer versions of Illustrator the Smart Guides have been augmented to include a "On Line" feature. Illustrator will snap an Anchor to an imaginary line with the same angle that extends that Anchor's associated Path segments.