Welcome to the forum.
The Frame Size for the Project will be what you chose, at New Project.
Now, you have several options:
You could create a New Project in 4:3, Import those Clips, and end up with a 4:3 output file.
Leave the black bars (Pillarbox).
Create a colored, or abstract background, so that the black bars are not so jarring. *
In the Widescreen 16:9 Project, use the Fixed Effect Motion>Scale to Scale them up to fill the width of the 16:9 Frame Size. However, this will effectly crop off top and bottom to accommodate (one can use the Fixed Effect Motion>Position to crop off pixels from either the top, or bottom). Note: and Scaling will diminish the Clip's quality, but it might be acceptable.
* If you do go the abstract background route, this article might be of help: http://forums.adobe.com/message/2471303#2471303
Actually, they are part of the clip.
This clip started out as a super8 home movie that a family member had scanned and placed on a blu-ray disk. The super8 is basically 4:3, but the video file that was given to me is 16:9. I guess the scanning company automatically defaulted to 16:9. So the black bars are now part of the file. OK, I can live with the black bars if I have to, but being slightly OCD, it bothers me that they are there.
OK, in the case of a poor telecine, you are correct - they ARE part of the Video.
Now, Crop WILL remove them, but as you have seen, what you get looks exactly the same, as there is nothing below, so it will just appear as black.
You could add a Color Matte, say a light beige, or unobtrusive color, below that Clip, and then when you Crop, you would see that.
The concept of the Abstract Background, is how news networks handle 4:3 Source Footage, in their Widescreen feeds. Some, like FoxNews, will Duplicate the 4:3 footage, on the Video Track below, then Scale that up to fill the 16:9 Widescreen Frame. A drop in quality is not meaningful, due to the next two steps. Then, they both Blur that duplicate, and "ghost" it back to about 10%. What one ends up with is the same Video playing "behind" the 4:3 material, but it's Blurred and also ghosted. A lot of the material that they receive, and especially from overseas IS 4:3, and that is how they handle it.
Good luck, and sorry that telecine house went 16:9 with 4:3 footage. However, if you Import and use that faux 16:9 footage in a 4:3 Project, the 4:3 Frame Size will effectively Crop out those black bars, but then you are in 4:3, which is not a bad thing. The choices here are really more aesthetic, and related to what you want the footage to look like, when you Export/Share/Publish it.
but being slightly OCD, it bothers me that they are there.
I completely understand. I follow my maids around with my bubble level, correcting every piece of artwork - at least I do not carry my level to my friends' houses, but when things are off, they bother me too. Spent a half-hour in my hotel room at the Four Seasons last week, trying to get all of the artwork straight...
Often, I live with black bars, but feel, just as you do, and that is why I wanted to give you a bunch of options to handle them. Many just let 'em go, but others (like me), want more and better, but there are limitations.