Andrea, The first thing to do is figure out why the receptacles have stopped working. If the receptacles themselves are bad or if the wiring at the back of the receptacles is bad, then there are probably other locations in the house where this is happening - or about to happen. Figure out the problem and then be sure to address all of the places that are likely to be affected. The National Electrical Code gives you several choices of what to do with the non-working receptacles.
You can replace the ungrounded receptacles with new ungrounded receptacles. This is fine for places where you only plan to plug in ungrounded things - vacuum cleaners, clock radios, lamps, many laptop computers, etc.
You can replace the ungrounded receptacles with grounding-type receptacles, leave them ungrounded, and protect them with GFCIs. In this case you have to label the receptacle faces, "No Equipment Ground." If the GFCI protection is from a remote location, you also have to label them, "GFCI Protected." This choice will protect people as well as grounding will but it won't protect electronic equipment because, as mentioned earlier, surge protectors won't work without a ground to shunt the surge to. You can replace the ungrounded receptacles with grounding-type receptacles and install a separate grounding conductor that runs from the new receptacle to the service panel or to a point on the grounding electrode system for the house. To do this properly is usually as much trouble as running a whole new cable. (You may *not* run a new grounding conductor to the "nearest water pipe" as used to be allowed in the days of yore.) Of course, you can also install an entirely new cable to the receptacle and abandon or, better yet, remove the old knob & tube wiring.Regarding the old knob & tube wiring, it's an obsolete system and you should at least consider replacing as much of it as you can. Always begin with the wiring in the attic because it's usually in the worst condition - particularly the wires directly above ceiling light fixtures, where the heat from the fixtures has burnt the insulation on the wires. The stuff in the attic and basement is easy to replace. Wiring in walls and in ceilings between floors is a little tougher but certainly possible. You can do a lot without disturbing much of the plaster walls if you learn how to use a 10' flexible drill bit. Thank you for the detailed response! I agree with the general consensus, replacing all the wiring would be ideal. However, that won't happen this year or even next. Or at all if we can't find an electrician that will preserve the walls. But for the short term, we would like to address the most pressing issues. Properly replacing the outlets that have gone out. Replacing wiring in basement since one of the breakers keeps tripping after lights are on for 2 mins, and a couple other small things.