I have purchased several of these small drawer units:
However, when I open the drawers, they come straight out - there is no mechanism to ensure that the drawers stay in the slot.
In the past, I have seen inexpensive drawer units use a small wooden peg glued into the top of the drawer cavity in order to keep the drawer from coming out - but these did not come with any such device.
Is there anything that I can install in order to keep these drawers from coming out?
You could consider drilling a small hole in the center of the back board of each drawer. Then drill a corresponding hole through the back side of the drawer cabinet unit itself. Remove the drawer and install a piece of heavy string through the hole in the back of the drawer. On the inside if the drawer knot the string so it cannot pull through the hole in the back of the drawer. (You could even tie the string to a small object like a large bead, or small washer). Make this string long enough that it can extend to inside the drawer cabinet when the drawer is fully removed.
Next reach into the back of the drawer cabinet and feed the other end of the string out through the hole in the back of the cabinet. Once pulled through the back hole then install the drawer into its slot and pull the excess string out through the rear of the cabinet. Now calibrate how much slack to leave in the string by pulling the drawer out as far as you think it should be allowed to go and still not pull free from its slot in the cabinet. Mark the string as to where it pulls back through the rear hole when the drawer is extended.
The next step is to secure the string at the marked position so that it cannot pull farther than the mark. This can be achieved in a number of ways as follows.
1) Knot the string at the mark with a large enough knot so it cannot pull back through the rear hole.
2) Tie a small object to the string at the mark position such as a large bead other object.
3) Pinch the end of the string under a washer and screw that is installed into the back if the drawer cabinet.
4) Fill the rear hole with some glue which would go in around the string and permanently affix it into the hole.
Looks like the drawers fit into the case w near zero clearance, so the standard peg through the side of the drawer thing won't work. A woodscrew into the case above the top of each drawer, and recessed a bit over 1 drawer-front thickness into the case should work. Put the screws in so they project down from the case 5-10 mm. You'll have to angle the drawers to get them in, and that could be tough if the screws are set too far back. Once the drawers are in, you can back out the screws a bit so the drawer-back is more securely captured. Tolerances are tight, so be careful not to put your screws all the way through the wood of the case.
Go to craft store and buy some velcro. Open up drawer halfway. Put velcro backing about an inch deep into drawer - I would use one piece on each side. After it is in place (they come with a glue side) then insert other side of velcro so it has some depth. Your drawer will get stuck when it hits it. Also you will be able to fully take the drawer out in the future - by removing the bottom velcro and some wiggling. Repeat for second drawer.
Remove the two drawers and turn the cabinet upside down so that the undersides of the 'roof panels' of the drawer pockets will be facing up.
In the middle, about 1/4" from the front edge, carve a tapered notch in each of these undersides. The notch should start about 1/8" deep near the front edge, and then taper to zero-depth towards the back.
Take a large paper clip and bend it open so the two U's are at about 100 degrees to each other.
On the inside-back of each drawer, attach one of these with the short 'U' up, the corner aligned flush with the top of the drawer back (or a little below it), such that the short 'U's will be pressing against the bottom side of the roofs and will relax upward into the carved notch as the draw is pulled out.
