Removing the hard disk from a VAIO VGN-FE31H is pretty easy. It’s located on the right hand side of the laptop, at the front (to the right of the touchpad).
If you look underneath the laptop the cover over the disk is easy to find. There are two screws in the cover; unscrew them, but don’t try to remove them as at least one of them has a washer underneath it that stops it being pulled out. Take off the cover leaving the screws in it.
Once the cover is removed, the disk can be seen, as in the picture below. When viewed from this angle, the SATA connectors are on the left hand side of the drive.
The drive is sitting in a metal tray, which is usually held in place by four screws. Two of these are the ones you unscrewed to take the cover off, and they go through the metal tabs on the right hand side. The other two are along the left hand edge of the drive. Take these out.
The drive tray can be removed by sliding it to the right. There’s a black plastic tab at the right hand side, which will probably be tucked into the gap between the two metal tabs. If you untuck this you can pull it to the right to slide the drive tray.
Once you’ve moved the tray to the right it will just lift up and you can remove it. There are four screws along the long edges of the disk. If these are taken out the disk can be completely removed from the drive bay.
To put the disk back, just do the reverse: put the tray back in its hole, and slide it to the left so the connectors plug in to the disk. Then tuck the plastic tab back into the gap between the metal tabs. Put the two screws back in on the left hand edge of the drive.
I’ve found that the black cover can be a bit of a pain to get back in. First insert the tabs on the left hand edge of the disk cover into the gaps in the case (two above the drive tray screws, and one more between these two). Then gently press the right hand side down. If the screws don’t seem to be tightening (especially the one towards the back of the laptop), gently push down on the edge of the cover a bit more and it should click into place.