You absolutely can do it. EASY. You don't need a hang on back filter either. I ran a 20 long around 20 years ago and kept soft corals, LPS and a clam and that setup today, not including the clam, was easily $300 or less.
I'm currently using a 20g for a frag tank and it's equipment would be $300 or less if I used a cheaper light.
You are going to have soft corals, so heavy flow isn't needed. One koralia nano 425 is probably fine or 2. Could do 2 koralia 240 instead. Back corners pointed forwards towards middle. $64
50w heater $20 - Eheim jager
Piece of live rock from your LFS to seed the tank $15 - find rest on craigslist. Or get 10 lbs from petsmart online $20
Aquatic life ro buddie 50g - $50
Light - find something used on craigslist. T5 great option. Should be able to find something for $50
Sand - caribsea arag alive special grade 20lb - $28 shipped
Total: $247
The only equipment above that I think is suspect is the aquatic life RO buddy. I've never used it and just found on Google. I ran a tank for years using tapwater filtered through a brita. So that RO should be better.