5km is considered a short distance run. Within a few weeks, it's something that should be achievable in sub-20/25 minutes.

In the UK, there's a volunteer led organisation called parkrun which hosts 5km runs/races in over 1000 parks around the country every Saturday. It's quite popular. My local race consistently get 300+ entrants each week. Worth checking out: https://www.parkrun.org.uk/

5km, 10km, half marathon and marathon are the main running distances. Less than 5km is generally stuff run on athletics tracks, which falls under athletics.