If an item is too powerful to repair, am I screwed?

You can repair the bow without using an anvil, but you'd lose the enchantments. I'm afraid that without mods you won't be able to repair your bow, use it wisely :-(

As fredley stated, you cannot repair it in game. But since you asked for ANY way to repair it, you could use NBTExplorer to set the damage of your bow to 0.

Simple open the level.dat file inside the folder of your world with NBTExplorer and klick Search->Find. Tick the two checkboxes and search for name:"id" and value:"261".

This will quickly show your bow, now you just have double-click "Damage" and enter 0.

That's it, your bow is saved ;) ( btw: that's an awesome bow! )

As of 1.9, you can get the Mending enchantment, which repairs it automatically while getting experience. Be careful though, this is a treasure enchantment, so you may only get it only through fishing, buying from villagers, or from chests in Generated Structures.

If you are willing to cheat, (And it's on single player) just click open to LAN and turn on cheats, then do "/gamemode 1" then you can repair it for free.