Items do indeed despawn, whether it be after a specific amount of time (reports have it at 20 minutes), or transitioning to a new day. Fortunately, you can still spawn the item back in using Creative Mode; you'll have to enable cheats for it, though, which disables Steam achievements.

To do so,
1. Boot up the game, load the world, and use /enablecheats in the in-game chat. This will enable cheats, granting you access to a wide variety of commands.
2. Use the /creative command to enter Creative Mode.
3. Open your inventory, click on the Creative tab on the left, and search for the bobblehead. Once found, drag it into your hotbar.
4. Disable Creative Mode via the same /creative command. You're done!

Unfortunately, your world now has cheats enabled, which disables Steam achievements.
If you don't care about them, you can continue with your newfound powers. There's lots of new commands to try!
If you do care about Steam achievements, though, and wish to collect as many as you can, you might be in luck. The original Windows-only version of the game has an Asset Manager that allows you to edit your world's save file to force-disable cheats.

While there is no chat command to disable cheats in-game, there is a way to disable them using the game's Asset Manager.
In your Steam library, right-click on the game and click on Properties. Head to the Local Files tab and click on Browse Local Files. This will bring you to Staxel's Steam folder. Open the bin directory and search for "Staxel.AssetManager.exe" - this is the program that allows you to mod the game!
Once open, allow it to load its asset file system. When it's ready, open the View tab and click on Database Editor. This will open the directory with all your worlds! Simply open your world's folder and search for the "universeSettings.bsdb" file. Once open, you'll be greeted by a window titled "Create Template". Click on the [...] button to the right of "(Collection)" and it'll bring up the Property Collection Editor. In the Members list, you'll see "clock" and "universe". Select "universe" and click on the arrow to the left of "Value" in the "universe properties" panel. Once again, click on the [...] button to the right of "(Collection)". This will open another Property Collection Editor. You should then select "cheated" in the Members list and its value will be displayed to the right. Simply click on the value and select False from the dropdown menu. Hit OK twice to find yourself back at the "Create Template" window. Click on "Save and Close" and open your world folder. Select "universeSettings.bsdb" and replace it.
You're done! Steam achievements should no longer be disabled for your world!
Huge thanks to Nimbus Fox, one of the community's avid modders, for figuring it out!

That's it! You can now complete your quest, relieved that the 404 Bobblehead Not Found error has been resolved.