Can I repair this corroded shutoff valve on my own?
That's a soldered valve, with a drain cap (typical for things that you might drain for the winter after shutting them off, though not always employed that way.) 

Replacement is not the only solution, and may not be the best option.

The packing nut leak (right side) might be as simple as using a pair of wrenches to slightly tighten the nut on the handle stem. The wrench close to the valve body is just holding still so you don't twist the pipes while tightening the nut closer to the handle. In more extreme cases you might need to repack it (shut off water, remove handle and nut, replace packing material.)

The drain cap (left side drip) may either need to be tightened slightly, or else shut off the valve, remove the cap, with a bucket to catch the water that drains out (if the valve was installed in the correct direction, you don't have to shut off the main supply) and replace the gasket in the cap, or get a new cap with gasket.