Just Angie's List some companies. It's a pretty painless process in terms of performing the work. If you're up to it, replacing a portion of the pipe yourself is pretty easy ASSUMING you can dig down to it. The County will own the main to your property line. There isn't a valve like a water line (where the liability would end) but there might be a Y for a cleanout. Either/or - once you get into your yard - it's on you. If not too deep and you know where the break is (you can have someone camera the line for you to pinpoint the break for a couple hundred bucks), simply dig it up. You will then need to cut the bad pipe out and remove it. Don't start flushing toilets/running water at this point in the house. If you are truely a novice, take a peice of the pipe to Home Depot and they will get you what you need. Cut a new peice of PVC and use the fittings HD/Lowes gave you and connect to the old pipe. Outside of that, the only thing you may need to fool with is some type of bedding/backfill for the pipe - but honestly, if you are just replacing 6' of pipe - I would just use what you excavated and put it back while packing it in place.
If you've backed up into the basement, "stuff" isn't passing the roots. Therefore you may want to investigate the idea of having a pumper truck available to suck out the line. If the line behind the clog is jammed up, when you free it, it will empty into your hole (and that can be both a mess and an environmental issue).
Is your basement higher than the street elevation wise? Unless you are on a pump, sewer falls on gravity. And since sewer pipe runs deep than water lines - the main is usually fairly deep (too deep to hand dig). But if your main leaves your house high enough and runs on a slop to the main - you may be able to catch it at a decent depth.
At the end of the day, I can't imagine it costing your $4k as noted above. Not saying someone didn't pay that, but, unless you are really deep, it's 1/2 days work with a backhoe to repair it. Materials cost virtually nothing.