You can do some polishing with an angle grinder and diamond pads ( see
www.braxtonbragg.com ). Just keep a bottle of water handy to keep the concrete wet.
HD has a dewalt grinder for about $40, might be cheap enough to just buy, but the pads are expensive.
4sq ft is not a lot, but it's not a little either. You couldn't do it with granite, but concrete is much softer so it might work. I don't have much exp with concrete.
BB also has grinding stones and cup wheels if you wanted to try that.
I would rule out doing it totally by hand.