For a project I have to cut an acrylic glass rod (diameter = 5mm) into a few small chunks. Then, I have to sand the cut surfaces and polish them. Since the surface to be polished is very small, I don't want to buy a special polishing agent. So is there a household substance that I can use as a substitute?
If you don't have a suitable abrasive substance nor wish to purchase some, you really have only toothpaste as a possible solution. Some products are not abrasive and will accomplish nothing. Some may be excessively abrasive and result in a poorer surface than desired. Obviously, testing is indicated if you try toothpaste.
On the other hand, flame polishing may provide what you require. A butane torch burns quite cleanly and can be controlled to the necessary extent. A gas range also burns cleanly, but control of such a large surface of burning propellant is going to be less easily managed.
Heat guns are another clean route, but if you have none, it's a purchase you wish to avoid.
Other heat sources are going to deposit soot or other impurities.
If you're able to locate a suitable heat source, experimentation is again indicated. It's easy to overheat and distort the plastic, especially 5 mm material. Aim for minimal application, as you can more easily repeat the heating than you can remove too much! Laser cut acrylic has nearly glass smooth edges, due to careful application of heat.
This image is of a 10 mm laser cut acrylic cube. Due to the high magnification, one can see the striations caused by the stepper motors. If a smooth surface was required, I'd sand it with the highest grit sandpaper I can find (1200) followed by micromesh (undesired purchase for you) with a water bath. The result would be glass smooth at the expense of some elbow grease. Using heat would be much faster and less laborious.