steel can generally be welded in most workshops, aluminum requires a specialist and carbon - forget it!
If you're talking about frames though, the foil-thin steel used in light performance frames takes unusual skill to weld. A neighbor of ours who has been welding for decades and has a list of NASA and aircraft certifications says he won't touch anything that thin. For carbon, just take some JB Weld. Carbon won't break from fatigue like steel will though. I and many riders I know have broken steel frames but not carbon even though we've crashed them many times. Our younger son, going 25mph, hit a car two summers ago that turned illegally in front of him. We sent the carbon frame and fork to Calfee for their thorough inspections and tests. They found no damage to the carbon itself. Now he still rides the same bike with a lot of new parts on it. One of our friends works at a place that makes carbon military helicopter blades. He says they have to be able to take 100 hits from a .50-caliber machine gun without failure, although they do have a lot of sensors in them and tell the pilot when enough damage says it's time to head for home. His father recently had a bad crash on his carbon bike, and before fixing it with JB Weld, he rode it for awhile with sharp, stiff carbon pieces 2-3" long sticking out of his right seat stay 1/8-1/4". The stuff is not as fragile as people have made it out to be.
For tires, if you carry some good booting material, you can repair almost any cut and keep riding until the tire is totally worn out, even if that takes thousands more miles. For small cuts and hole, you can boot with a 1.5" piece of tire liner with no sharp corners. There's no need to glue it to the inside of the tire. Just let the pressure of the tube hold it in place. For really huge cuts like nearly an inch long, use a 3"-long piece of an old tire with the beads cut off. Again, make sure there are no sharp edges on it. I carry such a scrap, but have never used it. There was a time I wish I had one, when we got a front blowout on a relatively new tandem tire. I had just read about the dollar-bill trick, so I used that (I think I used two bills, as the hole was about 3/4" long) and then inflated to just over half pressure and babied it the rest of the long, unsupported ride. American paper money is very strong, being half cotton and half linnen. Some other countries' paper money is nowhere near as strong and durable. I and my family have probably ridden tens of thousands of miles on tires with booted cuts up to nearly 1/4" though, including front tires, including the sidewall, and they have never been a problem. I have bought and maintained somewhere near 200 tires over the decades for myself and my family, and all
of the catastrophic, or nearly catastrophic, tire failures we've had were on new
or nearly new tires, usually with less than 300 miles on them, including one on the very first ride, where the carcass starts pulling apart from the inside, indicating a manufacturing defect. Now when I see and feel the bulge developing quickly, I know to let some pressure out and baby it all the way home, and then that tire goes in the trash, unlike ones that just get cuts and get booted.