Answer

Here, you have to use “dans“, there is no other choice.

How to know which to use

Caninonos‘s examples and explanation are perfectly right.

When you use “dedans“, you already know which container you are speaking of. When you use “dans“, you need to specify that container.

Voici une boîte, il y a des gants dedans.
Voici une boîte, il y a des gants dans.
Il y a des gants dedans cette boîte.
Il y a des gants dans cette boîte.

The difference is about the same as between “in” and “inside” (that is a rule of thumb, though, and I’m not sure if it’s always true).

Here’s a box, there are gloves inside.
There are gloves in this box.

In any case, you can never say “dedans quelque chose“.

Side note: translation advice

Original sentence:

My hands are in places that not a lot of other people’s go.

I would suggest:

Mes mains sont à des endroits où peu d’autres vont.

• “à des endroits“: this is almost an idiom, even if the place is enclosed.
• “peu d’autres (mains) vont“: well, if they’re not mine, they’re other people’s. The meaning is not changed, but the sentence is quite lighter.

As Random quite rightly highlighted, though a bit less elegant, the following is closer to the original idea:

Mes mains sont à des endroits où pas beaucoup d’autres vont.