I'm about to shoot a video with my DSLR, a NIKON D5000?

However, the built in microphone does not produce high quality sound recording. This camera model also does not allow external devices like a gun mic or radio mic to be connected because there's no 3.5mm jack. What other options do I have for me to record sound well?

You can't shoot video on any professional still camera because that is only for still pictures

Not having a microphone input is a fault with most DSLR's and it's important and something that would be an easy thing to add during manufacture, a glaring omission IMO.

All you can do is record any sound on another device, digital sound recorders are relatively cheap, I use an Olympus DS-40 for this which can accept any kind of microphone, but it does mean that recording the sound is another job you have to do when your already concentrating on the visuals, it makes a one man job effectively into a two man job particularly when recording conversations which I suppose it really is anyway. Syncing the two (audio and video) is fairly easy to do in any Video Editing software.

I usually leave the sound turned on in the camera to record 'ambient' noise, and use it's soundtrack along with the sound recorded by the DS-40, I can fade from one to the other easily.

Often the answer is to use pre-recorded soundtracks for ambient, thousands are available and mix that into the sound saved by the camera.