Originally Posted by grumpus

To remove the axle you'll need to press the bearing out on one side. Support the hub on a solid surface (large bench vice, concrete floor, tree stump) with a block of wood into which you have drilled a hole larger than the bearing but smaller than the flange, then tap the axle (start gently, then harder until the bearing moves) with a large hammer (avoid battering it with an undersized hammer) until the bearing comes out the bottom. The axle will then pull out of the bearing, or you may need to knock it out.

The problem you will have is identifying the axle in order to find a replacement, if there are no markings to help. Also the bearing may be damaged by the removal process. - press your thumb hard sideways against the inner race while turning it to feel for notches - compare with the bearing you didn't remove. Some bike shops will have a press for removing wheel bearings but many will use the hammer method. Alternatively you may be able to use a 2/3-leg gear puller to remove the axle but you'll need a steel plate instead of the wooden block, and space through the spokes for the puller arms.

To refit the bearing get a socket wrench the same size as the outer race and knock that with the hammer to drive the bearing into place - this avoids damaging the bearing, but you have to be careful it goes in straight.

But I have to say it may be easier to replace the hub - for that you only need a spoke wrench and a desire to learn how to build a wheel. Or just buy another wheel and keep this one to work on later. Maybe get a torque wrench too, so you get a better idea of how hard is too hard.