There are two major types of one-piece crank sets, differentiated by
their axle threading. The more common size uses 24 threads per inch,
but some bikes, particularly older U.S. made Schwinn and Mongoose
models, use 28 threads per inch. No part of the bearing assembly is
interchangeable between threadings.
First step is figure out which one you have and how long you want your new crank arm. If you don't have an old mongoose or schwinn it sounds like you probably have the 24 tpi. If you measure your crank, the size will be from the center of the bb spindle to the center of the pedal spindle. They seem to be about $15-25 online.
Minimal repair: Just get a replacement crank arm and do the repair yourself. I'd recommend getting some fresh grease too, so you'd probably end up spending $20-30 in parts and could probably do it yourself in an afternoon even if you've never done it before. From Sheldon Brown:
One-piece cranks are the easiest type to service, and require no special tools. All you need is a large adjustable wrench and a screwdriver.
Fresh Bearings repair: As long as you're replacing the crank arm, you might find it worth while to just spend an extra $10 for a new bottom bracket/bearings, especially if your bottom bracket is missing bearings or the surfaces they run against are deeply pitted. You also might want to spend another $10 or so for some new pedals if yours are loose or hard to spin (keep in mind most bike pedals don't fit one-piece cranks, so you'd have to be careful to get the right thread count/diameter). If the cups are stuck in the bottom bracket shell, however, this may require more tools.
Upgraded Components Repair: You can also get a $20 converter that will allow you to install a more modern cartridge bottom bracket with any regular square-taper crankset. You would need some more tools or to get a bike shop to do the work for you. You would also need about $25 for a crankset and $20 for a bottom bracket, $15 for new pedals (different standard). So something like this would be fairly involved and could easily be over $100 just for the parts.