Is it possible to fly using a small engine?
The short answer is yes you can use some cheaply available engines to achieve flight however you are presumably asking about things like small car engines or motorcycle engines etc. The issue you have here is that these engines are inherently different from a use case standpoint than your typical aircraft engine. Generally speaking, most car/motorcycle engines rev too high to be used in an aircraft, you actually want to lower the RPM, not raise it. You can read up on it here and here and here. You can pull it off with a stepper gear box (like a turbo prop uses) and it was even been done on a piston plane back in the 80's by Mooney. The other problem you will have with car engines is cooling them, generally cars are watercooled, something that adds a lot of weight and is not ideal for flight. A two stroke or other similar air cooled motorcycle/scooter engine may be a better thing to use. 

Many small engines will produce more than enough horsepower to achieve flight. Keep in mind the early piper cubs only had 40HP which can be easily found in many 2 stroke engines.  

Your best bet may be to look for a used VW bug engine as they are very similar to aircraft engines and even were used in some air frames at one point in history. Due to the cars high production numbers the engines are fairly easy to come by.

You did not list a jurisdiction in your question but the regulations and legislation on this change from place to place, so you may need to see what exactly you are allowed to do in your jurisdiction in relation to home built and experimental aircraft.